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split-medial-pigeonhole-programme .. 94d59ceaed

Author SHA1 Message Date
didericis 94d59ceaed Relabel medial comparison plots by walk depth 2026-06-15 02:11:27 -04:00
didericis 24af5485d2 Add medial cut comparison plots 2026-06-15 01:47:23 -04:00
didericis ea1ab0b986 Add source cap cut to medial tire figures 2026-06-15 01:11:04 -04:00
didericis c64c720e5a Draw the whole medial graph with all tire cuts 
Add a --whole mode to draw_medial_tire_cut.py that renders the entire
medial graph M(G) (the assembled cut graph), on a Kamada-Kawai layout,
with the recognised tires highlighted (black annular vertices, blue/red
teeth carrying walk depths, larger red bite apex) and the rest of M(G)
in grey. Add the resulting figure (Figure 3) and a describing paragraph
to the paper for the n=20 seed-72 example, via an \input-ed .tikz file.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-15 00:07:00 -04:00
didericis 9d7cb7644e Draw a medial tire cut from a random n=20 graph 
Add experiments/draw_medial_tire_cut.py, the paper-graphics companion
that imports run_experiment and emits a TikZ panel (walk-depth labels +
cut slits) per recognised tread via to_tikz. Add the resulting figure
(Example 3.2, Figure 2): the single recognised tread T_2 of the medial
tire decomposition of a random maximal planar graph on 20 vertices
(seed 72), an 8-cycle piece with a bite, labelled and cut.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-14 23:55:00 -04:00
didericis a22ca4b888 Add medial tire cut experiment and chaining section 
New experiments/run_medial_tire_cut_experiment.py: generates a random
maximal planar graph (stacked seed + random diagonal flips), builds the
medial graph, takes the tire decomposition at a random vertex level
source, walk-depth labels and cuts each full medial tire graph chained
down the tire tree, and assembles one final cut graph of M(G) with a
global label map (data only; graphics go in a separate script).

Fix label_and_cut: the root face is None, which collided with the
next(..., None) sentinel, leaving teeth unlabelled when the entry up
tooth lay inside a bite gap; use a distinct sentinel so the ascent to
the root face runs.

Add a "Chaining across the tire tree" section to the paper, clarifying
that the candidate parent down teeth are the boundary (singleton) down
teeth only -- bite teeth are interior to the parent and shared with no
child, so a lower-walk bite is skipped.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-14 23:46:49 -04:00
didericis b4ddc7da8b Add walk-depth labelling/cut script and worked example 
New experiments/medial_tire_cut_labelling.py: takes a full medial tire
graph and an entry up tooth and runs the walk-depth labelling-and-cut
procedure, reusing the full medial tire generator's model and emitting
TikZ. Add a generator-produced 8-tooth example to the paper (Figure 1,
Example 2.3) showing the labelling and the two cuts, plus a remark
fixing the cut's closing tooth for descended faces.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-14 22:00:52 -04:00
didericis 291f7e98c7 Add Medial Tire Cuts paper with walk-depth labelling and cut 
New paper "Medial Tire Cuts" citing the medial tire decompositions
paper. States the goal of decomposing the medial graph into a tree of
3-faces, and gives the walk-depth labelling-and-cut procedure for a
single full medial tire graph: a cut duplicates the annular vertex
where a face's tooth traversal closes (planar unzip), reducing the
inner faces to teeth.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-14 21:39:03 -04:00
21 changed files with 2755 additions and 1154 deletions
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papers/medial_pigeonhole_programme/paper.aux
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\relax
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-nested-tire-decompositions}
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A medial pigeonhole programme}}{1}{}\protected@file@percent }
\newlabel{def:medial-boundary-state}{{2.1}{2}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Kempe-cycle conservation across medial tires}}{2}{}\protected@file@percent }
\newlabel{lem:kempe-cycles}{{3.1}{2}}
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
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\bibcite{bauerfeld-medial-tire}{1}
\bibcite{bauerfeld-nested-tire-decompositions}{2}
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papers/medial_pigeonhole_programme/paper.fdb_latexmk
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%% filename: amsart-template.tex
%% American Mathematical Society
%% AMS-LaTeX v.2 template for use with amsart
%% ====================================================================
\documentclass{amsart}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{tikz}
\usetikzlibrary{backgrounds}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\begin{document}
\title{The Medial Pigeonhole Programme}
% author one information
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{}
\thanks{}
\subjclass[2010]{Primary }
\keywords{plane graph, triangulation, medial graph, tire graph, Tait coloring, Kempe chain, Four Colour Theorem}
\date{}
\dedicatory{}
\begin{abstract}
Building on the medial tire decomposition of a plane triangulation, we
formulate a pigeonhole programme for the Four Colour Theorem in medial
terms. Each tire carries a boundary-state restriction relation, and a
proper vertex $3$-colouring of the full medial graph is a compatible
selection of these boundary states across the tire tree. We state a
chain-pigeonhole conjecture asserting that the restriction relations
cannot remain mutually disjoint along every branch, and we refine the
boundary states by recording how two-colour Kempe cycles are routed
through each annular tire region. This yields a Kempe-enhanced
restriction relation and a notion of Kempe-compatible gluing along level
cycles.
\end{abstract}
\maketitle
\section{Introduction}
This paper continues the medial tire programme begun
in~\cite{bauerfeld-medial-tire}. We use freely the terminology and
notation introduced there. For a plane triangulation $G$ with fixed
embedding, $M(G)$ denotes the full medial graph, and the tire-tree
decomposition $\mathcal{T}(G,S)$ at a level source $S$
of~\cite{bauerfeld-nested-tire-decompositions} induces a decomposition
of $M(G)$ into full medial tire graphs $\mathsf{M}(T)$, one for each
tread $T$, glued along their boundary medial vertex sets
$\partial_{\mathrm{out}}\mathsf{M}(T)$ and
$\partial_{\mathrm{in}}\mathsf{M}(T)$. We also use the annular medial
cycle $A(T)$, its up and down teeth and their apexes, the bites and the
auxiliary plane graph $B(T)$, and the medial tire restriction relation
$R_T$ of~\cite{bauerfeld-medial-tire}.
By the Tait--medial correspondence of~\cite{bauerfeld-medial-tire},
proper vertex $3$-colourings of $M(G)$ are in natural bijection with
proper $3$-edge-colourings of the cubic planar dual $G^*$. Thus the
Four Colour Theorem is the assertion that the full medial graph of every
plane triangulation is properly vertex $3$-colourable, and the medial
tire decomposition turns this into a question about how local boundary
colourings compose across the tire tree.
\section{A medial pigeonhole programme}
The restriction relation $R_T$ records exactly the local information
needed to pass a medial $3$-colouring through a tire. In a nested
chain
\[
T_0 \supset T_1 \supset \cdots \supset T_k,
\]
the outer boundary state of $T_{i+1}$ must match an inner boundary
state allowed by $R_{T_i}$. Thus a proof of the Four Colour Theorem in
this framework would follow from a structural reason that these
restriction sets cannot remain mutually disjoint along every branch of
the tire tree.
\begin{definition}[Medial boundary state]
\label{def:medial-boundary-state}
A \emph{medial boundary state} on a boundary set
$\partial\mathsf{M}(T)$ is a proper vertex $3$-colouring of the
subgraph induced by that boundary set, considered up to permutation of
the three colours and the dihedral symmetries of the boundary walk
when that boundary is a cycle.
\end{definition}
\begin{conjecture}[Medial chain-pigeonhole principle]
\label{conj:medial-chain-pigeonhole}
There is a function $N(k)$ such that the following holds. Let
$T_0 \supset T_1 \supset \cdots \supset T_{N(k)}$ be a nested chain of
tire treads whose relevant boundary medial walks have length at most
$k$. Then two adjacent restriction relations in the chain have
compatible medial boundary states after colour permutation and boundary
symmetry. Equivalently, the chain contains a local gluing step that
cannot be obstructed by disjoint proper vertex $3$-colouring
restrictions.
\end{conjecture}
\begin{conjecture}[Medial tire route to the Four Colour Theorem]
\label{conj:medial-route-fct}
For every plane triangulation $G$ and every level source $S$, the
restriction relations $\{R_T : T \in V(\mathcal{T}(G,S))\}$ admit a
compatible selection of boundary states across the tire tree. Hence
$M(G)$ is properly vertex $3$-colourable, $G^*$ is properly
$3$-edge-colourable, and $G$ is properly $4$-vertex-colourable.
\end{conjecture}
\begin{remark}
Conjecture~\ref{conj:medial-route-fct} is equivalent in strength to
the Four Colour Theorem when combined with Tait's correspondence. The
point of the formulation is not to weaken the target theorem, but to
move the obstruction into finite boundary-state restrictions carried by
annular medial tire pieces.
\end{remark}
\section{Kempe-cycle conservation across medial tires}
We now record an additional structure carried by proper
$3$-colourings of medial graphs. This structure will be useful for
describing how colourings glue across level cycles.
Let $G$ be a plane triangulation and let $M=M(G)$ be its medial graph.
Let
\[
\varphi:V(M)\to\{1,2,3\}
\]
be a proper $3$-colouring of $M$. For a two-element colour set
$P=\{a,b\}\subseteq\{1,2,3\}$, let $M_P$ denote the subgraph of $M$
induced by the vertices of colours $a$ and $b$.
Since $M$ is $4$-regular and $\varphi$ is proper, every vertex of
$M_P$ has degree $2$ in $M_P$. Hence every component of $M_P$ is a
cycle. We call these components the $P$-Kempe cycles of $\varphi$.
\begin{lemma}[Kempe chains are cycles]
\label{lem:kempe-cycles}
Let $G$ be a plane triangulation, let $M=M(G)$, and let
$\varphi$ be a proper $3$-colouring of $M$. For each
$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$, every component of $M_P$ is a cycle.
\end{lemma}
\begin{proof}
Let $v\in V(M_P)$. In the medial graph $M$, the vertex $v$ has degree
$4$. Since $\varphi$ is a proper $3$-colouring, none of the neighbours
of $v$ has colour $\varphi(v)$. Thus all four neighbours of $v$ have
one of the two colours different from $\varphi(v)$.
In the medial graph of a plane triangulation, the neighbours of a
medial vertex occur in two opposite pairs corresponding to the two
faces incident with the corresponding edge of $G$. Around each such
triangular face, the three medial vertices receive all three colours.
Consequently, at $v$ there are exactly two neighbours of each colour
different from $\varphi(v)$. It follows that, in the subgraph induced
by any two colours $P$, every vertex has degree $2$. Hence each
component of $M_P$ is a cycle.
\end{proof}
Let $T$ be a medial tire region. We regard $T$ as an annular transition
region whose boundary consists of one outer level cycle and finitely
many inner level cycles:
\[
\partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
Here $C_0$ is the outer level cycle of $T$, and the cycles
$C_1,\ldots,C_m$ are the inner level cycles. Each inner level cycle
$C_i$ is also the outer level cycle of the corresponding child region
in the tire tree.
The following lemma is the basic conservation principle.
\begin{lemma}[Kempe-cycle conservation across level cycles]
\label{lem:kempe-conservation}
Let $C$ be a level cycle of $M$ separating a parent side from a child
side. Let $K$ be a $P$-Kempe cycle for some
$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$. Then $K$ cannot enter the child side
of $C$ without also leaving it.
Equivalently, the incidences of $K$ with $C$ are paired by the
components of $K$ lying on the child side of $C$, and also paired by the
components of $K$ lying on the parent side of $C$.
\end{lemma}
\begin{proof}
By the preceding lemma, $K$ is a cycle. The level cycle $C$ separates
the sphere into two closed regions, which we call the parent side and
the child side. Consider the intersection of $K$ with one of these
regions. Since $K$ is a cycle, no component of this intersection can
have exactly one boundary endpoint on $C$. Each component is either
closed within the region, or is a path with two boundary endpoints on
$C$. Thus every entrance through $C$ is paired with an exit through
$C$.
\end{proof}
We now use these Kempe cycles to single out the colourings of a full
medial tire graph that respect the annular tooth structure.
\begin{definition}[Kempe-balanced colouring]
\label{def:kempe-balanced}
Let $\varphi$ be a proper $3$-colouring of the full medial tire graph
$\mathsf{M}(T)$. For a colour pair $P=\{a,b\}$, let $\mathsf{M}(T)_P$ be
the subgraph induced by the vertices of colours $a$ and $b$. Since
$\mathsf{M}(T)$ need not be $4$-regular, the components of
$\mathsf{M}(T)_P$ are paths or cycles; we call them the $P$-\emph{Kempe
chains} of $\varphi$. Every vertex of colour $a$ or $b$ lies on exactly
one $P$-Kempe chain.
A \emph{valid face} is the outer face of $\mathsf{M}(T)$, or an interior
face of $B(T)$ that is not a tooth---namely the root face or a bite
inner-gap face in the sense of~\cite{bauerfeld-medial-tire}. The
\emph{tooth apexes incident to} a valid face $F$ are:
\begin{itemize}
\item the up-tooth apexes (\cite{bauerfeld-medial-tire}), when
$F$ is the outer face;
\item the singleton down-tooth apexes whose annular edge lies on $F$,
when $F$ is interior---the apex on annular edge $m$ being incident to
the innermost bite $(i,j)$ with $i<m<j$, or to the root face if there
is none.
\end{itemize}
Bite apexes are never incident to a valid face in this sense.
For a colour pair $P=\{a,b\}$ write $\nu_P(F)$ for the number of tooth
apexes incident to $F$ that are coloured $a$ or $b$---equivalently, that
lie on a $P$-Kempe chain. The colouring $\varphi$ is
\emph{Kempe-balanced} if $\nu_P(F)$ is even for every valid face $F$ and
every colour pair $P$.
\end{definition}
\begin{remark}[Necessity of Kempe-balance]
\label{rem:kempe-balance-necessary}
A proper $3$-colouring of $\mathsf{M}(T)$ can be part of a proper
$3$-colouring of the whole medial graph $M(G)$ only when it is
Kempe-balanced: if $\varphi$ is the restriction to $\mathsf{M}(T)$ of a
proper $3$-colouring of $M(G)$, then $\varphi$ is Kempe-balanced.
Equivalently, a colouring of $\mathsf{M}(T)$ that fails the parity
condition at some valid face and colour pair cannot extend to a proper
$3$-colouring of $M(G)$. This is an instance of Kempe-cycle
conservation (Lemma~\ref{lem:kempe-conservation}). The tooth apexes
incident to a valid face are boundary medial vertices
(\cite{bauerfeld-medial-tire}) lying on a single level
cycle of the tire decomposition: the up-tooth apexes lie on the outer
level cycle, and the singleton down-tooth apexes incident to an interior
non-tooth face lie on the inner level cycle bounding that face. In the
$4$-regular graph $M(G)$ each $P$-Kempe chain of $\mathsf{M}(T)$ closes
up into a $P$-Kempe cycle, which by Lemma~\ref{lem:kempe-conservation}
meets each level cycle in an even number of $P$-coloured incidences; for
a given valid face these incidences are exactly its incident tooth
apexes coloured $a$ or $b$, whence $\nu_P(F)$ is even.
This argument is verified computationally. For bite-free pieces---capped
triangulated annuli on annular cycles of length $6,8,10,12$---every proper
$3$-colouring of $M(G)$ restricts to a Kempe-balanced colouring. The same
holds for pieces carrying a bite, including the case where singleton down
teeth lie in the bite's inner-gap face: there the inner level cycle splits
into a child level cycle per gap, and conservation across each child cycle
supplies the parity (in the checked example the three singleton down apexes
of a bite gap are a rainbow in every restriction).
\end{remark}
More generally, let $T$ be a medial tire region with boundary
\[
\partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
For a $P$-Kempe cycle $K$, every component of $K\cap T$ is either a
cycle contained in $T$, or a path with two endpoints on
$\partial T$. Thus the $P$-Kempe arcs inside $T$ define a pairing of
the $P$-coloured boundary incidences of
\[
C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
This motivates the following refinement of boundary states.
\begin{definition}[Kempe-enhanced boundary state]
Let $T$ be a medial tire region with outer level cycle $C_0$ and inner
level cycles $C_1,\ldots,C_m$. Let
\[
\mathcal C(T)=C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
A \emph{Kempe-enhanced boundary state} on $T$ consists of the following
data:
\begin{enumerate}
\item a boundary colouring
\[
\alpha:V(\mathcal C(T))\to\{1,2,3\};
\]
\item for each colour pair
\[
P\in\{\{1,2\},\{2,3\},\{3,1\}\},
\]
a pairing $\pi_P$ of the $P$-coloured boundary incidences of
$\mathcal C(T)$ induced by the $P$-Kempe arcs lying inside $T$.
\end{enumerate}
We write such a state as
\[
\kappa=(\alpha,\pi_{12},\pi_{23},\pi_{31}).
\]
\end{definition}
Given a proper $3$-colouring $\varphi$ of the medial tire graph
$M(T)$, the restriction of $\varphi$ to the boundary level cycles gives
the boundary colouring $\alpha$, while the two-colour Kempe arcs inside
$T$ give the pairings $\pi_{12},\pi_{23},\pi_{31}$. Thus $\varphi$
determines a Kempe-enhanced boundary state, denoted
\[
\kappa_T(\varphi).
\]
\begin{definition}[Kempe-enhanced restriction relation]
The \emph{Kempe-enhanced restriction relation} of $T$ is
\[
\mathcal K_T
=
\left\{
\kappa_T(\varphi):
\varphi \text{ is a proper }3\text{-colouring of } M(T)
\right\}.
\]
This refines the ordinary boundary-colouring relation by recording not
only which boundary colourings extend across $T$, but also how the
two-colour Kempe cycles are routed through the annular tire region.
\end{definition}
The annular structure of a tire is useful in two distinct ways. First,
it gives a bounded transition region between level cycles: the colouring
of the annular medial cycle controls, and in many cases determines, the
colouring of the remaining medial tire vertices. Thus the number of
possible transition states is bounded in terms of the annular structure,
rather than the total size of the subtree below the tire. Second, it
describes how the outer level cycle and the inner level cycles are
related by Kempe arcs. The level cycles are the gluing interfaces, while
the annular tire is the transition operator between them.
\begin{definition}[Kempe-compatible gluing]
Let $T$ be a medial tire region and let $U$ be a child region glued to
$T$ along a common level cycle $C$. Thus $C$ is an inner level cycle of
$T$ and the outer level cycle of $U$.
Let
\[
\kappa_T=(\alpha_T,\pi^T_{12},\pi^T_{23},\pi^T_{31})
\in \mathcal K_T
\]
and
\[
\kappa_U=(\alpha_U,\pi^U_{12},\pi^U_{23},\pi^U_{31})
\in \mathcal K_U.
\]
We say that $\kappa_T$ and $\kappa_U$ are \emph{Kempe-compatible along
$C$} if:
\begin{enumerate}
\item the boundary colourings agree on $C$:
\[
\alpha_T|_{V(C)}=\alpha_U|_{V(C)};
\]
\item for each colour pair
\[
P\in\{\{1,2\},\{2,3\},\{3,1\}\},
\]
the pairings $\pi^T_P$ and $\pi^U_P$ compose along the
$P$-coloured incidences of $C$ without producing an unpaired endpoint.
\end{enumerate}
When these conditions hold, the composed pairings determine a
Kempe-enhanced boundary state on the exposed boundary of
$T\cup_C U$.
\end{definition}
In these terms, gluing local colourings is not merely a matter of
matching boundary colours. The colourings must also route their
two-colour Kempe arcs compatibly across every shared level cycle. The
ordinary restriction relation records whether a boundary colouring can
be extended locally; the Kempe-enhanced relation additionally records
the conservation of Kempe-cycle flow through the annular transition
region.
For a tire with one outer level cycle and several inner level cycles,
\[
\partial T=C_0\sqcup C_1\sqcup\cdots\sqcup C_m,
\]
the parent tire may correlate the boundary states on the different
inner cycles. The Kempe-enhanced relation records this correlation as
a system of pairings among the $P$-coloured incidences of all boundary
level cycles simultaneously. Thus one should view a medial tire as a
multi-output transition operator
\[
\mathcal K_T:
C_0 \leadsto (C_1,\ldots,C_m),
\]
rather than as an independent collection of binary transitions.
The guiding principle is therefore:
\begin{quote}
Level cycles are the interfaces used for gluing, while annular tire
regions are the bounded transition regions that route Kempe cycles
between those interfaces.
\end{quote}
\begin{thebibliography}{9}
\bibitem{bauerfeld-medial-tire}
E.~Bauerfeld,
\emph{Medial Tire Decompositions of Plane Triangulations},
manuscript (math-research repository), 2026.
\bibitem{bauerfeld-nested-tire-decompositions}
E.~Bauerfeld,
\emph{Nested Tire Decompositions of Plane Triangulations},
manuscript (math-research repository), 2026.
\bibitem{tait-original}
P.~G. Tait,
\emph{Remarks on the colourings of maps},
Proceedings of the Royal Society of Edinburgh \textbf{10} (1880),
729--729.
\end{thebibliography}
\end{document}
papers/medial_tire_cuts/experiments/draw_medial_tire_cut.py
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"""Draw the walk-depth labelling and cut of a medial tire decomposition.
Paper-graphics companion to ``run_medial_tire_cut_experiment.py``: it imports
``run_experiment`` from there, runs the pipeline on a random maximal planar
graph, and emits TikZ. By default it draws one ``tikzpicture`` (walk-depth
labels + cut slits) per recognised full medial tire graph, using ``to_tikz``
from ``medial_tire_cut_labelling``. With ``--whole`` it instead draws a
two-panel Figure 3 graphic: the source graph with its source highlighted and
the whole medial graph M(G) drawn with every medial vertex at the midpoint of
its source edge and labelled by that source edge, with the full BFS-level chain
shown and the currently computed walk-depth labels and cuts marked.
This script only renders; the experiment itself draws nothing. Run with the
repo venv (networkx): ``.venv/bin/python``.
Examples:
.venv/bin/python draw_medial_tire_cut.py -n 20 --seed 59 > panels.tex
.venv/bin/python draw_medial_tire_cut.py -n 20 --seed 59 --whole > whole.tex
"""
from __future__ import annotations
import argparse
import math
import os
import sys
import networkx as nx
import numpy as np
_HERE = os.path.dirname(os.path.abspath(__file__))
sys.path.insert(0, _HERE)
from run_medial_tire_cut_experiment import run_experiment # noqa: E402
from medial_tire_cut_labelling import to_tikz # noqa: E402
from tire_realization_analysis import triangular_faces # noqa: E402
def tikz_panels(n: int, seed: int, scale: float = 1.6,
min_degree: int = 5, attempts: int = 1000) -> tuple[dict, list[str]]:
"""Run the experiment and return ``(result, panels)``, one TikZ panel per
recognised tread, each showing that tread's walk-depth labelling and cut."""
result = run_experiment(n=n, seed=seed, min_degree=min_degree, attempts=attempts)
panels = []
for d in sorted(result["results"]):
rec = result["results"][d]
panels.append(to_tikz(rec["g"], depth=rec["depth"], cuts=rec["cuts"],
entry_edge=rec["entry_edge"], scale=scale))
return result, panels
# --------------------------------------------------------------------------- #
# Figure 3: the source graph and midpoint drawing of the whole medial graph.
# --------------------------------------------------------------------------- #
def _source_layout(G: nx.Graph) -> dict[int, tuple[float, float]]:
"""Straight-line planar layout for the source graph, normalised to the unit
box and reused by the medial drawing."""
faces, _ = triangular_faces(G)
outer = list(faces[0])
outer_set = set(outer)
raw = {}
for i, v in enumerate(outer):
angle = math.radians(90.0 - i * 360.0 / len(outer))
raw[v] = np.array([math.cos(angle), math.sin(angle)], dtype=float)
inner = [v for v in sorted(G.nodes()) if v not in outer_set]
if inner:
idx = {v: i for i, v in enumerate(inner)}
n = len(inner)
A = np.zeros((n, n))
bx = np.zeros(n)
by = np.zeros(n)
for i, v in enumerate(inner):
nbrs = list(G.neighbors(v))
A[i, i] = 1.0
for w in nbrs:
if w in idx:
A[i, idx[w]] -= 1.0 / len(nbrs)
else:
bx[i] += raw[w][0] / len(nbrs)
by[i] += raw[w][1] / len(nbrs)
xs = np.linalg.solve(A, bx)
ys = np.linalg.solve(A, by)
for v in inner:
raw[v] = np.array([xs[idx[v]], ys[idx[v]]], dtype=float)
pts = np.array([raw[v] for v in G.nodes()], dtype=float)
center = 0.5 * (pts.max(axis=0) + pts.min(axis=0))
span = float(max(*(pts.max(axis=0) - pts.min(axis=0)), 1.0))
return {
v: tuple((raw[v] - center) / span)
for v in G.nodes()
}
def _edge_midpoint(pos: dict, edge) -> tuple[float, float]:
u, v = edge
ux, uy = pos[u]
vx, vy = pos[v]
return (0.5 * (ux + vx), 0.5 * (uy + vy))
def _edge_label(edge) -> str:
u, v = edge
return f"${u}\\!{{-}}\\!{v}$"
def _source_graph_tikz(result: dict, pos: dict, scale: float) -> str:
G, source = result["G"], result["source"]
L = []
A = L.append
A(f"\\begin{{tikzpicture}}[scale={scale},")
A(" sedge/.style={black!50, line width=0.35pt},")
A(" sv/.style={circle, draw=black!60, fill=white, inner sep=1.1pt},")
A(" srcv/.style={circle, draw=blue!75!black, fill=blue!18, line width=0.7pt, inner sep=1.8pt}]")
def pt(v):
x, y = pos[v]
return f"({x:.3f},{y:.3f})"
for u, v in sorted(G.edges()):
A(f"\\draw[sedge] {pt(u)}--{pt(v)};")
for v in sorted(G.nodes()):
style = "srcv" if v == source else "sv"
A(f"\\node[{style}] at {pt(v)} {{}};")
sx, sy = pos[source]
A(f"\\node[font=\\scriptsize, text=blue!70!black] at ({sx:.3f},{sy - 0.085:.3f}) {{source {source}}};")
A("\\end{tikzpicture}")
return "\n".join(L)
def _medial_midpoint_tikz(result: dict, pos: dict, scale: float) -> str:
"""Draw M(G) with each medial vertex at the midpoint of its source edge.
Every medial vertex is labelled by its source edge; same-level source edges
show the BFS level-chain tooth layers, and interlevel source edges show the
annular layers. Currently computed tire walk-depth labels and cut labels
are overlaid without moving the medial vertices away from their source
edges."""
G, M = result["G"], result["M"]
levels = nx.single_source_shortest_path_length(G, result["source"])
medial_pos = {edge: _edge_midpoint(pos, edge) for edge in M.nodes()}
apex_roles = {}
apex_walks = {}
for r in result["labels"]:
apex_roles[r["apex"]] = r["role"]
apex_walks.setdefault(r["apex"], []).append(r["walk"])
cut_records = []
cut_number = 1
for c in result.get("cap_cuts", []):
cut_records.append((cut_number, c["medial_vertex"], "cap", c))
cut_number += 1
for d in sorted(result["results"]):
rec = result["results"][d]
g, bij = rec["g"], rec["bij"]
for c in rec["cuts"]:
if c.vertex is None:
continue
cut_records.append((cut_number, bij[f"a{c.vertex}"], d, c))
cut_number += 1
L = []
A = L.append
A(f"\\begin{{tikzpicture}}[scale={scale},")
A(" base/.style={black!12, line width=0.25pt},")
A(" med/.style={black!38, line width=0.32pt},")
A(" annv/.style={circle, draw=black!70, fill=black!18, inner sep=1.0pt},")
A(" levone/.style={circle, draw=orange!75!black, fill=orange!20, inner sep=1.2pt},")
A(" levtwo/.style={circle, draw=violet!70!black, fill=violet!18, inner sep=1.2pt},")
A(" levthree/.style={circle, draw=teal!70!black, fill=teal!18, inner sep=1.2pt},")
A(" knownv/.style={circle, draw=red!70!black, fill=red!24, inner sep=1.5pt},")
A(" elbl/.style={font=\\tiny, text=black!70, inner sep=0.2pt},")
A(" dlbl/.style={font=\\tiny\\bfseries, text=black, inner sep=0.5pt},")
A(" cut/.style={red!80!black, line width=1.0pt},")
A(" cutlbl/.style={font=\\tiny, text=red!75!black}]")
def pt_med(edge):
x, y = medial_pos[edge]
return f"({x:.3f},{y:.3f})"
def pt_src(v):
x, y = pos[v]
return f"({x:.3f},{y:.3f})"
for u, v in sorted(result["G"].edges()):
A(f"\\draw[base] {pt_src(u)}--{pt_src(v)};")
for u, v in M.edges():
A(f"\\draw[med] {pt_med(u)}--{pt_med(v)};")
def chain_style(edge):
u, v = edge
lu, lv = levels[u], levels[v]
if lu != lv:
return "annv"
if edge in apex_roles:
return "knownv"
return {1: "levone", 2: "levtwo", 3: "levthree"}.get(lu, "annv")
for mv in sorted(M.nodes()):
A(f"\\node[{chain_style(mv)}] at {pt_med(mv)} {{}};")
for mv in sorted(M.nodes()):
x, y = medial_pos[mv]
A(f"\\node[elbl] at ({x:.3f},{y:.3f}) [yshift=-4.8pt] {{{_edge_label(mv)}}};")
for mv in sorted(apex_walks):
x, y = medial_pos[mv]
label = ",".join(str(w) for w in sorted(apex_walks[mv]))
A(f"\\node[dlbl] at ({x:.3f},{y:.3f}) [yshift=5.0pt] {{{label}}};")
for number, mv, _d, _cut in cut_records:
u, v = mv
ux, uy = pos[u]
vx, vy = pos[v]
mx, my = medial_pos[mv]
ex, ey = vx - ux, vy - uy
length = math.hypot(ex, ey) or 1.0
dx, dy = -0.035 * ey / length, 0.035 * ex / length
A(f"\\draw[cut] ({mx - dx:.3f},{my - dy:.3f})--({mx + dx:.3f},{my + dy:.3f});")
A(f"\\node[cutlbl] at ({mx + 2.4 * dx:.3f},{my + 2.4 * dy:.3f}) {{cut {number}}};")
A("\\end{tikzpicture}")
return "\n".join(L)
def medial_tikz(result: dict, scale: float = 7.0) -> str:
"""Two-panel TikZ for Figure 3: the source graph and the midpoint drawing of
its medial graph with all medial vertices labelled, plus the tire
walk-depth labels and cuts."""
pos = _source_layout(result["G"])
source = _source_graph_tikz(result, pos, scale=0.58 * scale)
medial = _medial_midpoint_tikz(result, pos, scale=scale)
return "\n".join([
"\\begin{tabular}{c}",
source,
"\\\\[-0.25ex]",
"{\\scriptsize source graph $G$}",
"\\\\[1.0ex]",
medial,
"\\\\[-0.25ex]",
"{\\scriptsize medial graph $M(G)$ at edge midpoints}",
"\\end{tabular}",
])
def main() -> None:
parser = argparse.ArgumentParser(description=__doc__,
formatter_class=argparse.RawDescriptionHelpFormatter)
parser.add_argument("-n", type=int, default=20)
parser.add_argument("--seed", type=int, default=72)
parser.add_argument("--scale", type=float, default=1.6)
parser.add_argument("--min-degree", type=int, default=5,
help="reject random graphs below this minimum degree")
parser.add_argument("--attempts", type=int, default=1000,
help="number of consecutive seeds to try for --min-degree")
parser.add_argument("--whole", action="store_true",
help="draw the whole medial graph M(G) with all cuts, "
"instead of one panel per tread")
args = parser.parse_args()
if args.whole:
result = run_experiment(n=args.n, seed=args.seed,
min_degree=args.min_degree, attempts=args.attempts)
treads = sorted(result["results"])
print(f"% whole medial graph: n={args.n} seed={args.seed} "
f"graph_seed={result['graph_seed']} min_degree={result['min_degree']} "
f"source={result['source']} recognised treads={treads} "
f"|M(G)|={result['M'].number_of_nodes()}")
print(medial_tikz(result, scale=args.scale if args.scale != 1.6 else 7.0))
return
result, panels = tikz_panels(args.n, args.seed, scale=args.scale,
min_degree=args.min_degree, attempts=args.attempts)
treads = sorted(result["results"])
print(f"% medial tire cut: n={args.n} seed={args.seed} "
f"graph_seed={result['graph_seed']} min_degree={result['min_degree']} "
f"source={result['source']} recognised treads={treads}")
if not panels:
print("% (no recognised full medial tire graphs for this graph)")
for d, panel in zip(treads, panels):
g = result["results"][d]["g"]
print(f"% --- tread {d}: |A(T)|={g.n} word={g.tooth_word} "
f"bites={sorted(g.bites)} ---")
print(panel)
if __name__ == "__main__":
main()
papers/medial_tire_cuts/experiments/medial_tire_cut_labelling.py
+388
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"""Walk-depth labelling and cut of a full medial tire graph.
Implements the procedure of Definition 2.1 ("Walk-depth labelling and cut") of
the *Medial Tire Cuts* paper:
1. Pick an arbitrary up tooth, the entry tooth; it has walk depth d.
2. Traverse all teeth bounding the inner face incident to the entry tooth
clockwise until reaching the entry tooth, incrementing the walk depth by 1
for each tooth traversed.
3. On reaching the last tooth in the face, perform a cut by duplicating the
annular vertex at which the traversal closes (the annular vertex shared by
the last tooth and the closing tooth).
4. Find the tooth t of highest walk depth that is a member of a bite.
5. If t is incident to a face F with unlabelled teeth, traverse the teeth of F
starting from t in the direction of the unlabelled tooth incident to t
(sharing an annular vertex), incrementing the walk depth as you go.
6. Repeat steps 3-5 until all teeth are labelled.
The full medial tire graph model (annular cycle A(T), up/down teeth, bites, the
auxiliary plane graph B(T) and its inner faces) is the one from the companion
``full_medial_tire_generator.py`` of the medial tire decompositions paper, which
we import.
Teeth are identified with the annular edges that carry them: edge i sits on the
annular vertices a_i and a_{(i+1) mod n} and carries exactly one tooth. A bite
(i, j) carries two teeth, one on edge i and one on edge j, that share the bite
apex p. The inner non-tooth faces of B(T) are the root face (written ``None``)
and one inner-gap face per bite.
"""
from __future__ import annotations
import argparse
import math
import os
import sys
# Import the full medial tire model from the companion paper's experiments.
_GEN_DIR = os.path.normpath(os.path.join(
os.path.dirname(__file__), "..", "..",
"medial_tire_decompositions_of_plane_triangulations", "experiments",
))
sys.path.insert(0, _GEN_DIR)
from full_medial_tire_generator import ( # noqa: E402
FullMedialTireGraph,
has_incident_bite,
innermost_bite,
satisfies_bite_face_condition,
)
Face = "tuple[int, int] | None" # a bite (i, j), or None for the root face
# ---------------------------------------------------------------------------
# Face structure of B(T).
# ---------------------------------------------------------------------------
def parent_face(graph: FullMedialTireGraph, bite: tuple[int, int]) -> Face:
"""The face directly enclosing ``bite``: the minimal-span bite strictly
containing it, or the root face ``None``."""
i, j = bite
enclosing = [b for b in graph.bites if b[0] < i and b[1] > j]
if not enclosing:
return None
return min(enclosing, key=lambda b: b[1] - b[0])
def door_bite(graph: FullMedialTireGraph, edge: int) -> tuple[int, int] | None:
"""The bite that ``edge`` is a door of (i.e. a bite edge), or None."""
for b in graph.bites:
if edge in b:
return b
return None
def faces_bordered(graph: FullMedialTireGraph, edge: int) -> list[Face]:
"""The inner non-tooth faces whose boundary the tooth on ``edge`` lies on.
A bite door borders two faces (its bite's gap and that bite's parent); any
other tooth borders the single face directly containing its edge.
"""
bite = door_bite(graph, edge)
if bite is not None:
return [bite, parent_face(graph, bite)]
return [innermost_bite(edge, graph.bites)]
def face_boundary(graph: FullMedialTireGraph, face: Face) -> list[int]:
"""The teeth (annular edges) bounding ``face``, in clockwise cyclic order.
Clockwise is increasing edge index. For the root face the boundary is read
around the whole cycle; for a bite gap (i, j) it is read along the arc
i, i+1, ..., j and closes through the bite apex. Edges enclosed by a child
bite are skipped (they belong to the child's gap face).
"""
n = graph.n
arc = range(n) if face is None else range(face[0], face[1] + 1)
return [k for k in arc if face in faces_bordered(graph, k)]
def all_faces(graph: FullMedialTireGraph) -> list[Face]:
return [None] + sorted(graph.bites)
def shared_annular_vertex(graph: FullMedialTireGraph, e1: int, e2: int) -> int | None:
"""The annular vertex a_k shared by edges ``e1`` and ``e2``, or None."""
n = graph.n
common = {e1, (e1 + 1) % n} & {e2, (e2 + 1) % n}
return next(iter(common)) if common else None
# ---------------------------------------------------------------------------
# The walk-depth labelling and cut.
# ---------------------------------------------------------------------------
class Cut:
"""A cut performed when a face traversal closes: the duplicated annular
vertex, together with the last labelled tooth and the closing tooth that
share it, and the face being closed."""
__slots__ = ("vertex", "last_tooth", "closing_tooth", "face", "order")
def __init__(self, vertex, last_tooth, closing_tooth, face, order):
self.vertex = vertex
self.last_tooth = last_tooth
self.closing_tooth = closing_tooth
self.face = face
self.order = order
def __repr__(self):
f = "root" if self.face is None else f"bite{self.face}"
return (f"Cut(order={self.order}, a{self.vertex}, "
f"last=e{self.last_tooth}, closing=e{self.closing_tooth}, face={f})")
def label_and_cut(graph: FullMedialTireGraph, entry_edge: int,
start_depth: int = 0) -> tuple[dict[int, int], list[Cut]]:
"""Run the procedure starting from up tooth ``entry_edge``.
Returns ``(depth, cuts)`` where ``depth`` maps each annular edge (tooth) to
its walk depth, and ``cuts`` is the list of cuts in the order performed.
"""
if graph.tooth_word[entry_edge] != "U":
raise ValueError(f"entry edge {entry_edge} is not an up tooth")
depth: dict[int, int] = {}
cuts: list[Cut] = []
counter = start_depth
def traverse(face: Face, start_edge: int, is_entry: bool) -> None:
nonlocal counter
boundary = face_boundary(graph, face)
m = len(boundary)
pos = boundary.index(start_edge)
if is_entry:
depth[start_edge] = counter
counter += 1
direction = +1
else:
# head toward the unlabelled tooth incident to the door t
direction = +1 if boundary[(pos + 1) % m] not in depth else -1
last_new = start_edge
i = pos
while True:
i = (i + direction) % m
edge = boundary[i]
if edge in depth: # the closing tooth
cuts.append(Cut(
vertex=shared_annular_vertex(graph, last_new, edge),
last_tooth=last_new, closing_tooth=edge,
face=face, order=len(cuts),
))
return
depth[edge] = counter
counter += 1
last_new = edge
# Steps 1-3: the entry face.
traverse(innermost_bite(entry_edge, graph.bites), entry_edge, is_entry=True)
# Steps 4-6: descend (or ascend) through bites, deepest first. The root
# face is ``None``, so we use a distinct sentinel for "no unlabelled face".
_MISSING = object()
while len(depth) < graph.n:
labelled_bite_teeth = sorted(
(e for e in depth if door_bite(graph, e) is not None),
key=lambda e: depth[e], reverse=True,
)
for t in labelled_bite_teeth:
target = next((F for F in faces_bordered(graph, t)
if any(e not in depth for e in face_boundary(graph, F))),
_MISSING)
if target is not _MISSING:
traverse(target, t, is_entry=False)
break
else:
break # no progress possible
return depth, cuts
# ---------------------------------------------------------------------------
# TikZ rendering.
# ---------------------------------------------------------------------------
def _coords(graph: FullMedialTireGraph,
r_ann=1.0, r_up=1.46, r_down=0.60) -> dict[str, tuple[float, float]]:
n = graph.n
def ang(k): # a_0 at the top, increasing k clockwise
return math.radians(90.0 - k * 360.0 / n)
def edge_mid_dir(i): # angle of the bisector of edge i's two endpoints
a0, a1 = ang(i), ang((i + 1) % n)
return math.atan2(math.sin(a0) + math.sin(a1), math.cos(a0) + math.cos(a1))
pos = {f"a{k}": (r_ann * math.cos(ang(k)), r_ann * math.sin(ang(k)))
for k in range(n)}
for i in graph.up_edges:
a = edge_mid_dir(i)
pos[f"u{i}"] = (r_up * math.cos(a), r_up * math.sin(a))
for i in graph.singleton_down_edges:
a = edge_mid_dir(i)
pos[f"d{i}"] = (r_down * math.cos(a), r_down * math.sin(a))
for (i, j) in graph.bites:
pts = [pos[f"a{i}"], pos[f"a{(i + 1) % n}"],
pos[f"a{j}"], pos[f"a{(j + 1) % n}"]]
cx = sum(p[0] for p in pts) / 4.0
cy = sum(p[1] for p in pts) / 4.0
pos[f"p{i}_{j}"] = (0.9 * cx, 0.9 * cy)
return pos
def _edge_midpoint(pos, graph, edge):
n = graph.n
a, b = pos[f"a{edge}"], pos[f"a{(edge + 1) % n}"]
return (0.5 * (a[0] + b[0]), 0.5 * (a[1] + b[1]))
def to_tikz(graph: FullMedialTireGraph,
depth: dict[int, int] | None = None,
cuts: list[Cut] | None = None,
entry_edge: int | None = None,
scale: float = 2.2) -> str:
"""A standalone ``tikzpicture`` for ``graph``; if ``depth`` is given, draw
the walk-depth labels and (with ``cuts``) the cut marks."""
pos = _coords(graph)
n = graph.n
L = []
A = L.append
A(f"\\begin{{tikzpicture}}[scale={scale},")
A(" ann/.style={circle, fill=black, inner sep=1.0pt},")
A(" upv/.style={circle, draw=blue!70!black, fill=blue!12, inner sep=1.4pt},")
A(" downv/.style={circle, draw=red!70!black, fill=red!12, inner sep=1.4pt},")
A(" bitev/.style={circle, draw=red!70!black, fill=red!32, inner sep=1.7pt},")
A(" cyc/.style={black, line width=1.0pt},")
A(" tth/.style={black!55, line width=0.4pt},")
A(" lbl/.style={font=\\scriptsize},")
A(" dlbl/.style={font=\\scriptsize\\bfseries, text=black},")
A(" cut/.style={red!80!black, line width=1.3pt},")
A(" cutlbl/.style={font=\\tiny, text=red!75!black}]")
def pt(name):
x, y = pos[name]
return f"({x:.3f},{y:.3f})"
# annular cycle
cyc = "--".join(pt(f"a{k}") for k in range(n)) + "--cycle"
A(f"\\draw[cyc] {cyc};")
# spokes
for i in graph.up_edges:
A(f"\\draw[tth] {pt(f'u{i}')}--{pt(f'a{i}')} {pt(f'u{i}')}--{pt(f'a{(i+1)%n}')};")
for i in graph.singleton_down_edges:
A(f"\\draw[tth] {pt(f'd{i}')}--{pt(f'a{i}')} {pt(f'd{i}')}--{pt(f'a{(i+1)%n}')};")
for (i, j) in graph.bites:
apex = f"p{i}_{j}"
for e in (i, j):
A(f"\\draw[tth] {pt(apex)}--{pt(f'a{e}')} {pt(apex)}--{pt(f'a{(e+1)%n}')};")
# vertices
for k in range(n):
A(f"\\node[ann] at {pt(f'a{k}')} {{}};")
for i in graph.up_edges:
A(f"\\node[upv] at {pt(f'u{i}')} {{}};")
for i in graph.singleton_down_edges:
A(f"\\node[downv] at {pt(f'd{i}')} {{}};")
for (i, j) in sorted(graph.bites):
A(f"\\node[bitev] at {pt(f'p{i}_{j}')} {{}};")
# walk-depth labels: placed along the spoke from apex toward the edge mid
if depth is not None:
for edge in range(n):
apex = graph.apex_of_edge(edge)
ax, ay = pos[apex]
mx, my = _edge_midpoint(pos, graph, edge)
f = 0.5
lx, ly = ax + f * (mx - ax), ay + f * (my - ay)
A(f"\\node[dlbl] at ({lx:.3f},{ly:.3f}) {{{depth[edge]}}};")
# cut marks: a short red slit across the duplicated annular vertex
if cuts:
for c in cuts:
if c.vertex is None:
continue
vx, vy = pos[f"a{c.vertex}"]
rad = math.atan2(vy, vx)
dx, dy = 0.16 * math.cos(rad), 0.16 * math.sin(rad)
A(f"\\draw[cut] ({vx-dx:.3f},{vy-dy:.3f})--({vx+dx:.3f},{vy+dy:.3f});")
lx, ly = vx + 0.30 * math.cos(rad), vy + 0.30 * math.sin(rad)
A(f"\\node[cutlbl] at ({lx:.3f},{ly:.3f}) {{cut {c.order+1}}};")
if entry_edge is not None:
ex, ey = pos[graph.apex_of_edge(entry_edge)]
rad = math.atan2(ey, ex)
tx, ty = ex + 0.34 * math.cos(rad), ey + 0.34 * math.sin(rad)
A(f"\\node[lbl, text=blue!60!black] at ({tx:.3f},{ty:.3f}) {{entry}};")
A("\\end{tikzpicture}")
return "\n".join(L)
# ---------------------------------------------------------------------------
# Worked example and CLI.
# ---------------------------------------------------------------------------
def worked_example() -> FullMedialTireGraph:
"""A clean 8-tooth piece: one bite (0,4), three down singletons 1,2,3 in its
gap, three up teeth 5,6,7 in the root face."""
return FullMedialTireGraph(n=8, tooth_word="DDDDDUUU", bites=frozenset({(0, 4)}))
def _check(graph: FullMedialTireGraph) -> None:
assert not has_incident_bite(graph.bites, graph.n), "bite uses incident edges"
assert satisfies_bite_face_condition(graph.tooth_word, graph.bites), \
"violates the bite-face condition"
assert graph.tooth_word.count("U") >= 3, "fewer than three up teeth"
def _describe(graph, depth, cuts) -> str:
lines = ["edge type walk-depth"]
for e in range(graph.n):
t = graph.tooth_word[e]
kind = {"U": "up"}.get(t, "down")
if door_bite(graph, e) is not None:
kind = "bite"
lines.append(f" e{e} {kind:<5} {depth[e]}")
lines.append("cuts (in order):")
for c in cuts:
f = "root" if c.face is None else f"bite{c.face}"
lines.append(f" cut {c.order+1}: duplicate a{c.vertex} "
f"(closing tooth e{c.closing_tooth} of {f})")
return "\n".join(lines)
def main() -> None:
parser = argparse.ArgumentParser(description=__doc__,
formatter_class=argparse.RawDescriptionHelpFormatter)
parser.add_argument("--entry", default="u5",
help="entry up tooth, as an edge index or apex name like u5")
parser.add_argument("--start-depth", type=int, default=0)
parser.add_argument("--tikz", choices=["plain", "labelled", "both"],
help="emit TikZ for the worked example")
args = parser.parse_args()
entry = args.entry
edge = int(entry[1:]) if isinstance(entry, str) and entry.startswith("u") else int(entry)
graph = worked_example()
_check(graph)
depth, cuts = label_and_cut(graph, edge, start_depth=args.start_depth)
if args.tikz == "plain":
print(to_tikz(graph))
elif args.tikz == "labelled":
print(to_tikz(graph, depth=depth, cuts=cuts, entry_edge=edge))
elif args.tikz == "both":
print("% --- plain ---")
print(to_tikz(graph))
print("% --- labelled + cut ---")
print(to_tikz(graph, depth=depth, cuts=cuts, entry_edge=edge))
else:
print(f"worked example: n={graph.n} word={graph.tooth_word} "
f"bites={sorted(graph.bites)} entry=e{edge}")
print(_describe(graph, depth, cuts))
if __name__ == "__main__":
main()
papers/medial_tire_cuts/experiments/run_medial_tire_cut_experiment.py
+414
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@@ -0,0 +1,414 @@
"""Medial tire cut experiment.
End-to-end experiment for the *Medial Tire Cuts* paper:
1. Generate a random maximal planar graph G on n vertices (stacked seed plus
random diagonal flips; ``random_maximal_planar`` from the medial tire
decompositions experiments), optionally rejecting samples below a requested
minimum degree.
2. Build its medial graph M(G).
3. Take the nested tire decomposition at one random vertex level source: the
BFS-level treads, each realized as a FullMedialTireGraph.
4. Walk-depth label and cut each full medial tire graph, chaining the labels
down the tire tree, and assemble one final cut graph of M(G) with a global
label map.
This script produces *data* (the final cut graph and its labels); it draws
nothing. Anything for the paper (figures) lives in a separate script that
imports ``run_experiment`` from here.
Chaining rule (walk depths across the tire tree).
* The root tread (no recognised parent) is entered at an arbitrary up tooth
with walk depth 0.
* A child tread is entered at the up tooth whose apex is the *same medial
vertex* as the parent's down tooth of lowest walk depth -- a parent down
tooth and the child up tooth glued to it across the shared level cycle are
the same medial vertex of M(G). The entry up tooth's walk depth is that
parent down tooth's depth + 1, and the walk increments locally from there.
* The source cap contributes one additional cut. It is placed at the
counter-clockwise source edge incident to the cap down tooth whose apex is
the root tread's entry up tooth.
Run with the repo venv (networkx + scipy): ``.venv/bin/python``.
"""
from __future__ import annotations
import argparse
import json
import os
import random
import subprocess
import sys
import networkx as nx
# Reuse the realization pipeline from the medial tire decompositions paper, and
# the walk-depth labelling/cut from this paper's companion script.
_HERE = os.path.dirname(os.path.abspath(__file__))
_MTD = os.path.normpath(os.path.join(
_HERE, "..", "..",
"medial_tire_decompositions_of_plane_triangulations", "experiments"))
sys.path.insert(0, _MTD)
sys.path.insert(0, _HERE)
from tire_realization_analysis import ( # noqa: E402
ekey, extract_tread, medial_graph, medial_tire_facemodel,
random_maximal_planar, recognise, triangular_faces,
)
from medial_tire_cut_labelling import door_bite, label_and_cut # noqa: E402
# --------------------------------------------------------------------------- #
# 4. Walk-depth labelling and cut, chained down the tire tree.
# --------------------------------------------------------------------------- #
def _apex_vertex(g, bij, edge):
"""The medial vertex that is the apex of the tooth on annular ``edge``."""
return bij[g.apex_of_edge(edge)]
def _label_treads(treads, results):
"""Fill ``results[d]`` with the walk-depth labelling and cuts for each
recognised tread ``d``, chaining child entries to parent down teeth."""
for d in sorted(treads):
g, bij = treads[d]
parent = treads.get(d - 1)
if parent is None:
entry_edge, start_depth = g.up_edges[0], 0 # arbitrary root entry
else:
pg, pbij = parent
pdepth = results[d - 1]["depth"]
# parent down teeth, lowest walk depth first
down = sorted((pdepth[k], _apex_vertex(pg, pbij, k))
for k in pg.down_edges)
child_up_apex = {bij[f"u{m}"]: m for m in g.up_edges}
entry_edge = start_depth = None
for value, apex in down:
if apex in child_up_apex:
entry_edge, start_depth = child_up_apex[apex], value + 1
break
if entry_edge is None: # no shared apex (degenerate); root-style
entry_edge, start_depth = g.up_edges[0], 0
depth, cuts = label_and_cut(g, entry_edge, start_depth=start_depth)
results[d] = {"g": g, "bij": bij, "entry_edge": entry_edge,
"start_depth": start_depth, "depth": depth, "cuts": cuts}
def _cap_cut(G, emb, source, levels, results):
"""The source-cap cut determined by the first recognised tread's entry.
If the root tread enters at an up tooth whose apex is the level-1 edge
``xy``, then ``xy`` is a down tooth of the source cap. Cut the
counter-clockwise source edge incident to that down tooth. The returned
record also stores the local neighbour split used to unzip the medial
vertex in the whole medial graph.
"""
if not results:
return []
root_depth = min(results)
rec = results[root_depth]
g, bij = rec["g"], rec["bij"]
x, y = _apex_vertex(g, bij, rec["entry_edge"])
if levels.get(x) != 1 or levels.get(y) != 1:
return []
order = list(emb.neighbors_cw_order(source))
if x not in order or y not in order:
return []
next_cw = {v: order[(i + 1) % len(order)] for i, v in enumerate(order)}
prev_cw = {v: order[(i - 1) % len(order)] for i, v in enumerate(order)}
if next_cw[x] == y:
cut_endpoint, other_endpoint = x, y
elif next_cw[y] == x:
cut_endpoint, other_endpoint = y, x
else:
return []
other_cap_endpoint = prev_cw[cut_endpoint]
mv = ekey(source, cut_endpoint)
return [{
"medial_vertex": mv,
"down_tooth": ekey(cut_endpoint, other_endpoint),
"neighbours_a": [
ekey(source, other_endpoint),
ekey(cut_endpoint, other_endpoint),
],
"neighbours_b": [
ekey(source, other_cap_endpoint),
ekey(cut_endpoint, other_cap_endpoint),
],
}]
# --------------------------------------------------------------------------- #
# Assemble one final cut graph of M(G) with a global label map.
# --------------------------------------------------------------------------- #
def _assemble_cut_graph(M, results, cap_cuts=None):
"""Apply every tread's cuts to M(G).
Each cut duplicates an annular medial vertex, splitting its four incident
medial edges along the slit between the two teeth meeting there: the tooth
on the previous annular edge (with that edge's far annular vertex) goes to
one copy, the tooth on the next annular edge to the other.
Returns ``(H, label_records, warnings)`` where ``H`` is the cut graph (a
networkx graph whose split vertices are keyed ``(medial_vertex, "A"/"B",
tread)``) and ``label_records`` lists every tooth's walk depth.
"""
# Per cut annular vertex: map each original neighbour -> which copy keeps it.
split = {} # medial_vertex -> {neighbour_medial_vertex: copy_node}
warnings = []
for i, c in enumerate(cap_cuts or []):
mv = c["medial_vertex"]
if mv in split:
warnings.append(f"cap cut at {mv} was already cut; skipped")
continue
copy_a = (mv, "A", f"cap{i}")
copy_b = (mv, "B", f"cap{i}")
split[mv] = {
**{v: copy_a for v in c["neighbours_a"]},
**{v: copy_b for v in c["neighbours_b"]},
}
for d in sorted(results):
g, bij = results[d]["g"], results[d]["bij"]
n = g.n
for c in results[d]["cuts"]:
kk = c.vertex
if kk is None:
continue
mv = bij[f"a{kk}"]
if mv in split:
warnings.append(f"annular vertex a{kk} of tread {d} cut twice; "
f"second cut not applied")
continue
e_prev, e_next = (kk - 1) % n, kk
copy_a = (mv, "A", d)
copy_b = (mv, "B", d)
split[mv] = {
bij[f"a{(kk - 1) % n}"]: copy_a,
_apex_vertex(g, bij, e_prev): copy_a,
bij[f"a{(kk + 1) % n}"]: copy_b,
_apex_vertex(g, bij, e_next): copy_b,
}
def resolve(node, other):
return split[node][other] if node in split else node
H = nx.Graph()
H.add_nodes_from(v for v in M.nodes() if v not in split)
for v, copies in split.items():
H.add_nodes_from(set(copies.values()))
for u, v in M.edges():
H.add_edge(resolve(u, v), resolve(v, u))
label_records = []
for d in sorted(results):
g, bij, depth = results[d]["g"], results[d]["bij"], results[d]["depth"]
for k in range(g.n):
role = ("up" if g.tooth_word[k] == "U"
else "bite" if door_bite(g, k) is not None else "down")
label_records.append({
"tread": d, "edge": k, "role": role,
"apex": _apex_vertex(g, bij, k), "walk": depth[k],
})
return H, label_records, warnings
# --------------------------------------------------------------------------- #
# Driver.
# --------------------------------------------------------------------------- #
def random_maximal_planar_min_degree(n: int, seed: int, flips: int = 400,
min_degree: int = 0,
attempts: int = 1000) -> tuple[nx.Graph, int]:
"""Generate a random maximal planar graph with minimum degree at least
``min_degree``. The returned seed is the actual sample seed used."""
if min_degree <= 0:
return random_maximal_planar(n, seed, flips=flips), seed
if min_degree >= 5:
plantri = os.path.normpath(os.path.join(_HERE, "..", "..", "..",
"plantri", "plantri"))
if os.path.exists(plantri):
data = subprocess.check_output(
[plantri, f"-m{min_degree}", "-g", str(n)],
stderr=subprocess.DEVNULL)
graphs = [line for line in data.splitlines() if line]
if graphs:
G = nx.from_graph6_bytes(graphs[seed % len(graphs)])
return nx.convert_node_labels_to_integers(G), seed
for offset in range(attempts):
sample_seed = seed + offset
G = random_maximal_planar(n, sample_seed, flips=flips)
if min(dict(G.degree()).values()) >= min_degree:
return G, sample_seed
raise RuntimeError(
f"no random maximal planar graph on {n} vertices with "
f"minimum degree >= {min_degree} found in {attempts} attempts "
f"starting at seed {seed}")
def run_experiment(n: int = 12, seed: int = 0, flips: int = 400,
min_degree: int = 5, attempts: int = 1000) -> dict:
"""Run the full pipeline and return a structured result.
Result keys: ``n, seed, G, M, source, treads`` (dict depth -> (g, bij)),
``results`` (dict depth -> labelling/cut record), ``skipped`` (list of
(depth, reason)), ``cut_graph`` (networkx graph), ``labels`` (list of tooth
records), ``warnings``.
"""
G, graph_seed = random_maximal_planar_min_degree(
n, seed, flips=flips, min_degree=min_degree, attempts=attempts)
faces, emb = triangular_faces(G)
M = medial_graph(G)
source = random.Random(f"source-{graph_seed}").choice(sorted(G.nodes()))
levels = nx.single_source_shortest_path_length(G, source)
treads, skipped = {}, []
for d in range(max(levels.values())):
tread = extract_tread(faces, levels, d)
if tread is None:
skipped.append((d, "no tread faces"))
continue
if len(tread["up"]) < 3:
skipped.append((d, f"only {len(tread['up'])} up teeth"))
continue
rec = recognise(medial_tire_facemodel(tread["tread_faces"]), tread)
if rec is None:
skipped.append((d, "not a valid full medial tire graph"))
continue
treads[d] = rec
results = {}
_label_treads(treads, results)
cap_cuts = _cap_cut(G, emb, source, levels, results)
cut_graph, labels, warnings = _assemble_cut_graph(M, results, cap_cuts=cap_cuts)
return {
"n": n, "seed": seed, "graph_seed": graph_seed,
"min_degree": min(dict(G.degree()).values()),
"G": G, "M": M, "source": source,
"treads": treads, "results": results, "cap_cuts": cap_cuts,
"skipped": skipped,
"cut_graph": cut_graph, "labels": labels, "warnings": warnings,
}
# --------------------------------------------------------------------------- #
# Serialization / reporting.
# --------------------------------------------------------------------------- #
def _vname(v) -> str:
"""Stable string name for a medial vertex (an edge key) or a split node."""
if isinstance(v, tuple) and len(v) == 3 and v[1] in ("A", "B"):
mv, side, d = v
return f"{mv[0]}-{mv[1]}#{side}@T{d}"
return f"{v[0]}-{v[1]}"
def to_json(result: dict) -> dict:
res = result["results"]
treads_out = []
for d in sorted(res):
g, bij = res[d]["g"], res[d]["bij"]
depth, cuts = res[d]["depth"], res[d]["cuts"]
teeth = [{
"edge": k,
"role": ("up" if g.tooth_word[k] == "U"
else "bite" if door_bite(g, k) is not None else "down"),
"apex": _vname(_apex_vertex(g, bij, k)),
"walk": depth[k],
} for k in range(g.n)]
treads_out.append({
"depth": d, "n": g.n, "tooth_word": g.tooth_word,
"bites": sorted(list(b) for b in g.bites),
"entry_edge": res[d]["entry_edge"], "start_depth": res[d]["start_depth"],
"teeth": teeth,
"cuts": [{
"annular_index": c.vertex,
"annular_vertex": _vname(bij[f"a{c.vertex}"]),
"last_edge": c.last_tooth, "closing_edge": c.closing_tooth,
} for c in cuts],
})
H = result["cut_graph"]
return {
"n": result["n"], "seed": result["seed"],
"graph_seed": result["graph_seed"], "min_degree": result["min_degree"],
"source": result["source"],
"graph_edges": sorted([int(u), int(v)] for u, v in result["G"].edges()),
"medial_vertices": result["M"].number_of_nodes(),
"skipped": [[d, why] for d, why in result["skipped"]],
"cap_cuts": [{
"medial_vertex": _vname(c["medial_vertex"]),
"down_tooth": _vname(c["down_tooth"]),
} for c in result["cap_cuts"]],
"treads": treads_out,
"cut_graph": {
"nodes": sorted(_vname(v) for v in H.nodes()),
"edges": sorted([_vname(u), _vname(v)] for u, v in H.edges()),
},
"labels": [{
"tread": r["tread"], "edge": r["edge"], "role": r["role"],
"apex": _vname(r["apex"]), "walk": r["walk"],
} for r in result["labels"]],
"warnings": result["warnings"],
}
def summary(result: dict) -> str:
H, res = result["cut_graph"], result["results"]
lines = [
f"random maximal planar graph: n={result['n']} requested seed={result['seed']} "
f"graph seed={result['graph_seed']} "
f"({result['G'].number_of_edges()} edges, min degree {result['min_degree']})",
f"medial graph M(G): {result['M'].number_of_nodes()} vertices",
f"level source: vertex {result['source']}",
f"recognised treads: {sorted(res)}",
f"skipped treads: {result['skipped']}",
]
for d in sorted(res):
g = res[d]["g"]
ncuts = len(res[d]["cuts"])
lines.append(
f" tread {d}: |A(T)|={g.n} word={g.tooth_word} "
f"bites={sorted(g.bites)} entry=e{res[d]['entry_edge']} "
f"start_depth={res[d]['start_depth']} cuts={ncuts}")
lines.append(
f"final cut graph: {H.number_of_nodes()} vertices, "
f"{H.number_of_edges()} edges, "
f"{len(result['cap_cuts']) + sum(len(r['cuts']) for r in res.values())} cuts total")
if result["warnings"]:
lines.append("warnings: " + "; ".join(result["warnings"]))
return "\n".join(lines)
def main() -> None:
parser = argparse.ArgumentParser(
description=__doc__, formatter_class=argparse.RawDescriptionHelpFormatter)
parser.add_argument("-n", type=int, default=12, help="number of vertices")
parser.add_argument("--seed", type=int, default=0)
parser.add_argument("--flips", type=int, default=400,
help="number of random diagonal flips when building G")
parser.add_argument("--min-degree", type=int, default=5,
help="reject random graphs below this minimum degree")
parser.add_argument("--attempts", type=int, default=1000,
help="number of consecutive seeds to try for --min-degree")
parser.add_argument("--json", metavar="PATH",
help="write the full result as JSON to PATH")
args = parser.parse_args()
result = run_experiment(n=args.n, seed=args.seed, flips=args.flips,
min_degree=args.min_degree, attempts=args.attempts)
print(summary(result))
if args.json:
with open(args.json, "w") as fh:
json.dump(to_json(result), fh, indent=2)
print(f"wrote {args.json}")
if __name__ == "__main__":
main()
papers/medial_tire_cuts/paper.aux
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\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Cutting a full medial tire graph}}{1}{}\protected@file@percent }
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\citation{bauerfeld-medial-tire}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The recognised tread $T_2$ of the medial tire decomposition of a random maximal planar graph on $20$ vertices (Example\nonbreakingspace 3.2\hbox {}), with its walk-depth labelling and cut. Black vertices are the annular medial vertices of $A(T)$; blue vertices are up-tooth apexes and red vertices down-tooth apexes, the larger red vertex being the shared apex of the bite on annular edges $2$ and $5$. Each tooth carries its walk depth; the red slits are the two cuts.}}{4}{}\protected@file@percent }
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\bibcite{bauerfeld-medial-tire}{1}
\newlabel{tocindent-1}{0pt}
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The source graph $G$ and the whole medial graph $M(G)$ of the minimum-degree-$5$ maximal planar graph on $20$ vertices generated by \texttt {plantri -m5} at seed $59$. The source vertex $5$ is highlighted in the top panel. In the bottom panel, each medial vertex is placed at the midpoint of its corresponding source edge and labelled by that edge. Black vertices come from source edges between consecutive levels; coloured vertices come from source edges within a single level of the chain. The red-highlighted vertices, walk-depth labels, and seven red slits are the computed source-cap cut and full-medial-tire labelling cuts for the recognised treads $T_1$ and $T_2$. Drawn by \texttt {experiments/draw\_medial\_tire\_cut.py} with \texttt {--whole --min-degree 5}.}}{5}{}\protected@file@percent }
\newlabel{fig:whole-medial}{{3}{5}}
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\begin{document}
\title{Medial Tire Cuts}
% author one information
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{}
\thanks{}
\subjclass[2010]{Primary }
\keywords{plane graph, triangulation, medial graph, tire graph, Tait coloring, Four Colour Theorem}
\date{}
\dedicatory{}
\begin{abstract}
Starting from the medial tire decomposition of a plane triangulation, we
study the cuts that medial tires make in the full medial graph. We will
show how to use medial tires to decompose the medial graph into a tree of
three faces.
\end{abstract}
\maketitle
\section{Introduction}
This paper builds on the medial tire decomposition
of~\cite{bauerfeld-medial-tire}. For a plane triangulation $G$ with
fixed embedding we use freely the terminology and notation introduced
there: the full medial graph $M(G)$, its decomposition into full medial
tire graphs $\mathsf{M}(T)$ indexed by the treads $T$ of the tire tree
$\mathcal{T}(G,S)$ at a level source $S$, the annular medial cycle
$A(T)$, and the boundary medial vertex sets.
We will show how to use medial tires to decompose the medial graph into
a tree of three faces.
\section{Cutting a full medial tire graph}
We first describe a procedure that simultaneously \emph{labels} and
\emph{cuts} a single full medial tire graph $\mathsf{M}(T)$ so that,
after the cuts, the only faces are the outer face and $3$-faces
(triangles)---the teeth of~\cite{bauerfeld-medial-tire}. The labelling
assigns to each tooth an integer \emph{walk depth}; the cuts break the
cyclic adjacencies of the teeth so that what remains is a tree of
$3$-faces.
By a \emph{cut} we mean the duplication of a single vertex of
$\mathsf{M}(T)$: the vertex is split into two copies and the embedding is
slit open along it (a planar unzip), separating the faces that meet only
at that vertex. A cut therefore reduces the number of bounded faces that
are not teeth.
Throughout we use the teeth, up and down teeth, apexes, bites, the
annular medial cycle $A(T)$, and the auxiliary plane graph $B(T)$
of~\cite{bauerfeld-medial-tire}. Each tooth is a $3$-face of
$\mathsf{M}(T)$, and the inner faces of $B(T)$ (the root face and the
bite inner-gap faces) are the larger faces to be cut into teeth.
\begin{definition}[Walk-depth labelling and cut]
\label{def:walk-depth-cut}
Let $\mathsf{M}(T)$ be a full medial tire graph. Assign walk depths and
cuts as follows.
\begin{enumerate}
\item Pick an arbitrary up tooth, the \emph{entry tooth}. It has walk
depth $d$.
\item Traverse all the teeth that bound the inner face incident to the
entry tooth clockwise until we reach the entry tooth, incrementing the
walk depth by $1$ for each tooth traversed. (The \emph{inner face
incident to the entry tooth} is the inner face of $B(T)$ whose boundary
contains the annular edge of $A(T)$ carrying the entry tooth.)
\item When you reach the last tooth in the face, perform a \emph{cut}
by duplicating the annular vertex at which the traversal closes---the
annular vertex of $A(T)$ shared by the last tooth and the entry tooth.
\item Find the tooth $t$ with the highest walk depth which is a member
of a bite.
\item If $t$ is incident to a face $F$ with unlabelled teeth, traverse
the teeth in $F$ starting from $t$ in the direction of the tooth
incident to $t$ which is unlabelled, and increment the walk depth by
$1$ as you travel. (Here a tooth is \emph{incident to $t$} when it
shares an annular vertex of $A(T)$ with $t$.)
\item Repeat steps (3)--(5) until all teeth have been labelled.
\end{enumerate}
\end{definition}
\begin{remark}[Closing tooth of a descended face]
\label{rem:closing-tooth}
For the entry face the traversal of step (2) closes by returning to the
entry tooth, so the cut of step (3) duplicates the annular vertex shared
by the last tooth and the entry tooth. For a face $F$ entered in step
(5), the traversal instead closes upon reaching an already-labelled
tooth: the other tooth of the bite through which $F$ was entered. In
both cases the cut of step (3) duplicates the annular vertex shared by
the last newly labelled tooth and this \emph{closing tooth}. Since both
teeth of a bite are labelled while traversing its parent face, every
descended face closes on such a tooth.
\end{remark}
\begin{example}[A worked walk-depth labelling and cut]
\label{ex:worked-cut}
Figure~\ref{fig:worked-cut} shows a full medial tire graph with annular
cycle of length $8$, generated by the full medial tire generator
of~\cite{bauerfeld-medial-tire}. Its eight teeth are: three up teeth on
the annular edges $5,6,7$ in the root face; one bite pairing the annular
edges $0$ and $4$; and three singleton down teeth on the annular edges
$1,2,3$ lying in that bite's inner-gap face.
Take the up tooth on edge $5$ as the entry tooth, with walk depth $0$.
Its inner face is the root face, bounded by the teeth on edges
$5,6,7,0,4$ in clockwise order. Step (2) labels them
\[
5\mapsto 0,\quad 6\mapsto 1,\quad 7\mapsto 2,\quad
0\mapsto 3,\quad 4\mapsto 4,
\]
and step (3) cuts by duplicating the annular vertex $a_5$ shared by the
last tooth (edge $4$) and the entry tooth (edge $5$). The highest-depth
bite tooth is now the one on edge $4$ (walk depth $4$); it is incident to
the still-unlabelled inner-gap face of the bite $(0,4)$. Entering that
face from edge $4$ toward its unlabelled neighbour, step (5) labels the
three down teeth
\[
3\mapsto 5,\quad 2\mapsto 6,\quad 1\mapsto 7,
\]
and closes on the already-labelled bite tooth of edge $0$, so step (3)
cuts by duplicating the annular vertex $a_1$
(Remark~\ref{rem:closing-tooth}). All eight teeth are now labelled, and
the two cuts leave only the outer face and the eight teeth as
$3$-faces. The labelling and cuts are produced by the script
\texttt{experiments/medial\_tire\_cut\_labelling.py}.
\end{example}
\begin{figure}[h]
\centering
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\qquad
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\caption{A full medial tire graph (left) and its walk-depth labelling and
cut (right), from Example~\ref{ex:worked-cut}. Black vertices are the
annular medial vertices of the cycle $A(T)$; blue vertices are up-tooth
apexes, red vertices are down-tooth apexes, and the larger red vertex is
the shared apex of the bite on annular edges $0$ and $4$. On the right,
each tooth carries its walk depth, and the two red slits mark the cuts:
\emph{cut~1} duplicates $a_5$ as the root-face traversal closes, and
\emph{cut~2} duplicates $a_1$ as the bite's inner-gap face closes. After
the cuts the only bounded faces are the eight teeth.}
\label{fig:worked-cut}
\end{figure}
\section{Chaining across the tire tree}
Definition~\ref{def:walk-depth-cut} labels and cuts a single full medial
tire graph. We extend it to the whole medial graph $M(G)$ through the
medial tire decomposition of~\cite{bauerfeld-medial-tire}: the tire tree
decomposes $M(G)$ into full medial tire graphs $\mathsf{M}(T)$, one per
tread $T$, glued along their boundary medial vertices. A parent tread's
inner level cycle is a child tread's outer level cycle, and the boundary
medial vertices on that shared cycle belong to both treads.
The key incidence is this. A \emph{boundary} (singleton) down tooth of a
parent tread and the up tooth of the child tread glued to it across the
shared level cycle have the \emph{same apex}: both apexes are the same
medial vertex of $M(G)$, namely the medial vertex of an edge with both
endpoints on the shared level cycle. We use this to carry the walk depth
from a parent into its children.
We label tread by tread, outward from the root:
\begin{itemize}
\item a tread with no parent in the decomposition---in particular the
innermost recognised tread---is treated as a \emph{root} and entered at
an arbitrary up tooth with walk depth $0$;
\item a child tread is entered at the up tooth whose apex is the parent's
boundary down tooth of lowest walk depth; that entry up tooth's walk depth
is one more than that down tooth's, and the walk then increments locally
within the child as in Definition~\ref{def:walk-depth-cut}.
\end{itemize}
The source cap contributes one additional cut before the recognised
treads are assembled. If the root tread enters at an up tooth whose apex
is the cap down tooth $xy$, we cut the cap annular vertex corresponding
to the counter-clockwise source edge incident to $xy$. In the example of
Figure~\ref{fig:whole-medial}, the root entry apex is the cap down tooth
$14\!-\!4$, so the cap cut is placed at the medial vertex $14\!-\!5$.
\begin{remark}[Candidate down teeth for chaining]
\label{rem:chaining-candidates}
The down teeth eligible to fix a child's entry are exactly the
\emph{boundary} (singleton) down teeth of the parent: those lying in a
single tread face, whose apex is the shared boundary medial vertex glued to
a child up tooth. A bite's two down teeth are \emph{not} eligible. By the
definition of a bite in~\cite{bauerfeld-medial-tire} its annular edge borders
two tread faces, so a bite tooth is interior to the parent tread and its
apex is a boundary medial vertex of no child. Hence ``the down tooth of
lowest walk depth'' is read among the boundary down teeth only; a bite of
even lower walk depth is skipped.
\end{remark}
Applying every tread's cuts to $M(G)$ assembles the per-tread labellings
and cuts into a single cut graph of $M(G)$ together with a global
walk-depth label map. This pipeline---random maximal planar graph, medial
graph, tire decomposition at a vertex level source, and chained walk-depth
labelling and cut---is carried out by the experiment script
\texttt{experiments/run\_medial\_tire\_cut\_experiment.py}.
\begin{example}[A medial tire cut from a random graph]
\label{ex:real-cut}
Run on a random maximal planar graph on $20$ vertices (seed $72$, level
source vertex $9$), the experiment yields a single recognised tread
$T_2$, drawn in Figure~\ref{fig:real-cut} with the walk-depth labelling
and cut emitted by the graphics companion
\texttt{experiments/draw\_medial\_tire\_cut.py}. Its annular cycle has
length $8$, with up teeth on annular edges $0,3,4$, singleton down teeth
on $1,6,7$, and a bite on the non-incident annular edges $2$ and $5$ (the
central shared apex). Entering at the up tooth on edge $0$ with walk
depth $0$, the root face is labelled in order ($0,1,2$ then $3,4,5$) and
\emph{cut~1} duplicates $a_0$ as it closes; the walk then descends through
the bite into its inner-gap face, labelling the two teeth there ($6,7$),
and \emph{cut~2} duplicates $a_3$ as that face closes. The two cuts leave
only the outer face and the eight teeth as $3$-faces.
\end{example}
\begin{figure}[h]
\centering
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\caption{The recognised tread $T_2$ of the medial tire decomposition of a
random maximal planar graph on $20$ vertices
(Example~\ref{ex:real-cut}), with its walk-depth labelling and cut. Black
vertices are the annular medial vertices of $A(T)$; blue vertices are
up-tooth apexes and red vertices down-tooth apexes, the larger red vertex
being the shared apex of the bite on annular edges $2$ and $5$. Each
tooth carries its walk depth; the red slits are the two cuts.}
\label{fig:real-cut}
\end{figure}
Figure~\ref{fig:whole-medial} repeats the whole-medial-graph drawing on a
random maximal planar graph on $20$ vertices with minimum degree $5$
(plantri seed $59$, level source vertex $5$). The experiment recognises
two full medial tire treads, $T_1$ and $T_2$, and produces seven cuts:
one source-cap cut and six full-tread cuts. The
top panel shows the source triangulation with its level source
highlighted; the bottom panel draws $M(G)$ on the same straight-line
embedding by placing each medial vertex at the midpoint of its
corresponding source edge. Every medial vertex is labelled by that source
edge. Black vertices correspond to source edges joining consecutive
levels, and coloured vertices correspond to source edges within one level.
The red-highlighted vertices, walk-depth labels, and red slits are the
computed full-medial-tire labelling and cuts.
\begin{figure}[h]
\centering
\input{whole_medial_seed59_min5.tikz}
\caption{The source graph $G$ and the whole medial graph $M(G)$ of the
minimum-degree-$5$ maximal planar graph on $20$ vertices generated by
\texttt{plantri -m5} at seed $59$. The source vertex $5$ is highlighted
in the top panel. In the bottom panel, each medial vertex is placed at
the midpoint of its corresponding source edge and labelled by that edge.
Black vertices come from source edges between consecutive levels; coloured
vertices come from source edges within a single level of the chain. The
red-highlighted vertices, walk-depth labels, and seven red slits are the
computed source-cap cut and full-medial-tire labelling cuts for the
recognised treads $T_1$ and $T_2$. Drawn by
\texttt{experiments/draw\_medial\_tire\_cut.py} with
\texttt{--whole --min-degree 5}.}
\label{fig:whole-medial}
\end{figure}
\begin{thebibliography}{9}
\bibitem{bauerfeld-medial-tire}
E.~Bauerfeld,
\emph{Medial Tire Decompositions of Plane Triangulations},
manuscript (math-research repository), 2026.
\end{thebibliography}
\end{document}
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% whole medial graph: n=20 seed=59 graph_seed=59 min_degree=5 source=5 recognised treads=[1, 2] |M(G)|=54
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\end{tikzpicture}
\\[-0.25ex]
{\scriptsize medial graph $M(G)$ at edge midpoints}
\end{tabular}
papers/medial_tire_cuts/whole_medial_seed72.tikz
+382
View File
@@ -0,0 +1,382 @@
% whole medial graph: n=20 seed=72 source=9 recognised treads=[2] |M(G)|=54
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\\[-0.25ex]
{\scriptsize source graph $G$}
\\[1.0ex]
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papers/medial_tire_decompositions_of_plane_triangulations/paper.aux
+17 -10
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@@ -1,5 +1,4 @@
\relax
\citation{bauerfeld-medial-pigeonhole}
\citation{bauerfeld-nested-tire-decompositions}
\citation{bauerfeld-nested-tire-decompositions}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
@@ -14,7 +13,7 @@
\newlabel{thm:annular-medial-colour-bound}{{3.3}{3}}
\newlabel{def:annular-teeth}{{3.4}{3}}
\newlabel{rem:teeth-sharing}{{3.5}{3}}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces A full medial tire graph $\mathsf {M}(T)$ illustrating the tooth terminology. The thick cycle is the annular medial cycle $A(T)$, whose black vertices are the annular medial vertices. Each edge of $A(T)$ carries one tooth: up teeth (blue apexes, outer-boundary medial vertices) point into the outer region, and down teeth (red apexes, inner-boundary medial vertices) point into the inner region. The two down teeth meeting at the central shared apex (larger red vertex) form a bite; that shared apex splits the inner region into two faces, one with four down teeth on its boundary and one with none.}}{4}{}\protected@file@percent }
@@ -27,17 +26,25 @@
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papers/medial_tire_decompositions_of_plane_triangulations/paper.tex
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@@ -53,12 +53,9 @@ isomorphic to the medial graph of the planar dual $G^*$, and proper
$3$-vertex-colourings of $M(G)$ are equivalent to proper
$3$-edge-colourings of the cubic dual. Thus Tait's reformulation of
the Four Colour Theorem may be studied through proper vertex
$3$-colourings of medial subgraphs. We define medial tire pieces and
their boundary-state restriction relations, and show that a proper
vertex $3$-colouring of $M(G)$ amounts to a compatible selection of
these boundary states across the tire tree. The resulting
pigeonhole programme for the Four Colour Theorem is developed
in~\cite{bauerfeld-medial-pigeonhole}.
$3$-colourings of medial subgraphs. We define medial tire pieces,
their boundary-state restriction relations, and a chain-pigeonhole
conjecture for compatible medial boundary states across the tire tree.
\end{abstract}
\maketitle
@@ -659,6 +656,340 @@ properness is already enforced by one of the local colourings. Hence
$\varphi$ is a proper vertex $3$-colouring of $M(G)$.
\end{proof}
\section{A medial pigeonhole programme}
The restriction relation $R_T$ records exactly the local information
needed to pass a medial $3$-colouring through a tire. In a nested
chain
\[
T_0 \supset T_1 \supset \cdots \supset T_k,
\]
the outer boundary state of $T_{i+1}$ must match an inner boundary
state allowed by $R_{T_i}$. Thus a proof of the Four Colour Theorem in
this framework would follow from a structural reason that these
restriction sets cannot remain mutually disjoint along every branch of
the tire tree.
\begin{definition}[Medial boundary state]
\label{def:medial-boundary-state}
A \emph{medial boundary state} on a boundary set
$\partial\mathsf{M}(T)$ is a proper vertex $3$-colouring of the
subgraph induced by that boundary set, considered up to permutation of
the three colours and the dihedral symmetries of the boundary walk
when that boundary is a cycle.
\end{definition}
\begin{conjecture}[Medial chain-pigeonhole principle]
\label{conj:medial-chain-pigeonhole}
There is a function $N(k)$ such that the following holds. Let
$T_0 \supset T_1 \supset \cdots \supset T_{N(k)}$ be a nested chain of
tire treads whose relevant boundary medial walks have length at most
$k$. Then two adjacent restriction relations in the chain have
compatible medial boundary states after colour permutation and boundary
symmetry. Equivalently, the chain contains a local gluing step that
cannot be obstructed by disjoint proper vertex $3$-colouring
restrictions.
\end{conjecture}
\begin{conjecture}[Medial tire route to the Four Colour Theorem]
\label{conj:medial-route-fct}
For every plane triangulation $G$ and every level source $S$, the
restriction relations $\{R_T : T \in V(\mathcal{T}(G,S))\}$ admit a
compatible selection of boundary states across the tire tree. Hence
$M(G)$ is properly vertex $3$-colourable, $G^*$ is properly
$3$-edge-colourable, and $G$ is properly $4$-vertex-colourable.
\end{conjecture}
\begin{remark}
Conjecture~\ref{conj:medial-route-fct} is equivalent in strength to
the Four Colour Theorem when combined with Tait's correspondence. The
point of the formulation is not to weaken the target theorem, but to
move the obstruction into finite boundary-state restrictions carried by
annular medial tire pieces.
\end{remark}
\subsection{Kempe-cycle conservation across medial tires}
We now record an additional structure carried by proper
$3$-colourings of medial graphs. This structure will be useful for
describing how colourings glue across level cycles.
Let $G$ be a plane triangulation and let $M=M(G)$ be its medial graph.
Let
\[
\varphi:V(M)\to\{1,2,3\}
\]
be a proper $3$-colouring of $M$. For a two-element colour set
$P=\{a,b\}\subseteq\{1,2,3\}$, let $M_P$ denote the subgraph of $M$
induced by the vertices of colours $a$ and $b$.
Since $M$ is $4$-regular and $\varphi$ is proper, every vertex of
$M_P$ has degree $2$ in $M_P$. Hence every component of $M_P$ is a
cycle. We call these components the $P$-Kempe cycles of $\varphi$.
\begin{lemma}[Kempe chains are cycles]
\label{lem:kempe-cycles}
Let $G$ be a plane triangulation, let $M=M(G)$, and let
$\varphi$ be a proper $3$-colouring of $M$. For each
$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$, every component of $M_P$ is a cycle.
\end{lemma}
\begin{proof}
Let $v\in V(M_P)$. In the medial graph $M$, the vertex $v$ has degree
$4$. Since $\varphi$ is a proper $3$-colouring, none of the neighbours
of $v$ has colour $\varphi(v)$. Thus all four neighbours of $v$ have
one of the two colours different from $\varphi(v)$.
In the medial graph of a plane triangulation, the neighbours of a
medial vertex occur in two opposite pairs corresponding to the two
faces incident with the corresponding edge of $G$. Around each such
triangular face, the three medial vertices receive all three colours.
Consequently, at $v$ there are exactly two neighbours of each colour
different from $\varphi(v)$. It follows that, in the subgraph induced
by any two colours $P$, every vertex has degree $2$. Hence each
component of $M_P$ is a cycle.
\end{proof}
Let $T$ be a medial tire region. We regard $T$ as an annular transition
region whose boundary consists of one outer level cycle and finitely
many inner level cycles:
\[
\partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
Here $C_0$ is the outer level cycle of $T$, and the cycles
$C_1,\ldots,C_m$ are the inner level cycles. Each inner level cycle
$C_i$ is also the outer level cycle of the corresponding child region
in the tire tree.
The following lemma is the basic conservation principle.
\begin{lemma}[Kempe-cycle conservation across level cycles]
\label{lem:kempe-conservation}
Let $C$ be a level cycle of $M$ separating a parent side from a child
side. Let $K$ be a $P$-Kempe cycle for some
$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$. Then $K$ cannot enter the child side
of $C$ without also leaving it.
Equivalently, the incidences of $K$ with $C$ are paired by the
components of $K$ lying on the child side of $C$, and also paired by the
components of $K$ lying on the parent side of $C$.
\end{lemma}
\begin{proof}
By the preceding lemma, $K$ is a cycle. The level cycle $C$ separates
the sphere into two closed regions, which we call the parent side and
the child side. Consider the intersection of $K$ with one of these
regions. Since $K$ is a cycle, no component of this intersection can
have exactly one boundary endpoint on $C$. Each component is either
closed within the region, or is a path with two boundary endpoints on
$C$. Thus every entrance through $C$ is paired with an exit through
$C$.
\end{proof}
We now use these Kempe cycles to single out the colourings of a full
medial tire graph that respect the annular tooth structure.
\begin{definition}[Kempe-balanced colouring]
\label{def:kempe-balanced}
Let $\varphi$ be a proper $3$-colouring of the full medial tire graph
$\mathsf{M}(T)$. For a colour pair $P=\{a,b\}$, let $\mathsf{M}(T)_P$ be
the subgraph induced by the vertices of colours $a$ and $b$. Since
$\mathsf{M}(T)$ need not be $4$-regular, the components of
$\mathsf{M}(T)_P$ are paths or cycles; we call them the $P$-\emph{Kempe
chains} of $\varphi$. Every vertex of colour $a$ or $b$ lies on exactly
one $P$-Kempe chain.
A \emph{valid face} is the outer face of $\mathsf{M}(T)$, or an interior
face of $B(T)$ that is not a tooth---namely the root face or a bite
inner-gap face of Remark~\ref{rem:bite-face-count}. The \emph{tooth
apexes incident to} a valid face $F$ are:
\begin{itemize}
\item the up-tooth apexes (Definition~\ref{def:annular-teeth}), when
$F$ is the outer face;
\item the singleton down-tooth apexes whose annular edge lies on $F$,
when $F$ is interior---the apex on annular edge $m$ being incident to
the innermost bite $(i,j)$ with $i<m<j$, or to the root face if there
is none.
\end{itemize}
Bite apexes are never incident to a valid face in this sense.
For a colour pair $P=\{a,b\}$ write $\nu_P(F)$ for the number of tooth
apexes incident to $F$ that are coloured $a$ or $b$---equivalently, that
lie on a $P$-Kempe chain. The colouring $\varphi$ is
\emph{Kempe-balanced} if $\nu_P(F)$ is even for every valid face $F$ and
every colour pair $P$.
\end{definition}
\begin{remark}[Necessity of Kempe-balance]
\label{rem:kempe-balance-necessary}
A proper $3$-colouring of $\mathsf{M}(T)$ can be part of a proper
$3$-colouring of the whole medial graph $M(G)$ only when it is
Kempe-balanced: if $\varphi$ is the restriction to $\mathsf{M}(T)$ of a
proper $3$-colouring of $M(G)$, then $\varphi$ is Kempe-balanced.
Equivalently, a colouring of $\mathsf{M}(T)$ that fails the parity
condition at some valid face and colour pair cannot extend to a proper
$3$-colouring of $M(G)$. This is an instance of Kempe-cycle
conservation (Lemma~\ref{lem:kempe-conservation}). The tooth apexes
incident to a valid face are boundary medial vertices
(Definition~\ref{def:boundary-medial-vertices}) lying on a single level
cycle of the tire decomposition: the up-tooth apexes lie on the outer
level cycle, and the singleton down-tooth apexes incident to an interior
non-tooth face lie on the inner level cycle bounding that face. In the
$4$-regular graph $M(G)$ each $P$-Kempe chain of $\mathsf{M}(T)$ closes
up into a $P$-Kempe cycle, which by Lemma~\ref{lem:kempe-conservation}
meets each level cycle in an even number of $P$-coloured incidences; for
a given valid face these incidences are exactly its incident tooth
apexes coloured $a$ or $b$, whence $\nu_P(F)$ is even.
This argument is verified computationally. For bite-free pieces---capped
triangulated annuli on annular cycles of length $6,8,10,12$---every proper
$3$-colouring of $M(G)$ restricts to a Kempe-balanced colouring. The same
holds for pieces carrying a bite, including the case where singleton down
teeth lie in the bite's inner-gap face: there the inner level cycle splits
into a child level cycle per gap, and conservation across each child cycle
supplies the parity (in the checked example the three singleton down apexes
of a bite gap are a rainbow in every restriction).
\end{remark}
More generally, let $T$ be a medial tire region with boundary
\[
\partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
For a $P$-Kempe cycle $K$, every component of $K\cap T$ is either a
cycle contained in $T$, or a path with two endpoints on
$\partial T$. Thus the $P$-Kempe arcs inside $T$ define a pairing of
the $P$-coloured boundary incidences of
\[
C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
This motivates the following refinement of boundary states.
\begin{definition}[Kempe-enhanced boundary state]
Let $T$ be a medial tire region with outer level cycle $C_0$ and inner
level cycles $C_1,\ldots,C_m$. Let
\[
\mathcal C(T)=C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
A \emph{Kempe-enhanced boundary state} on $T$ consists of the following
data:
\begin{enumerate}
\item a boundary colouring
\[
\alpha:V(\mathcal C(T))\to\{1,2,3\};
\]
\item for each colour pair
\[
P\in\{\{1,2\},\{2,3\},\{3,1\}\},
\]
a pairing $\pi_P$ of the $P$-coloured boundary incidences of
$\mathcal C(T)$ induced by the $P$-Kempe arcs lying inside $T$.
\end{enumerate}
We write such a state as
\[
\kappa=(\alpha,\pi_{12},\pi_{23},\pi_{31}).
\]
\end{definition}
Given a proper $3$-colouring $\varphi$ of the medial tire graph
$M(T)$, the restriction of $\varphi$ to the boundary level cycles gives
the boundary colouring $\alpha$, while the two-colour Kempe arcs inside
$T$ give the pairings $\pi_{12},\pi_{23},\pi_{31}$. Thus $\varphi$
determines a Kempe-enhanced boundary state, denoted
\[
\kappa_T(\varphi).
\]
\begin{definition}[Kempe-enhanced restriction relation]
The \emph{Kempe-enhanced restriction relation} of $T$ is
\[
\mathcal K_T
=
\left\{
\kappa_T(\varphi):
\varphi \text{ is a proper }3\text{-colouring of } M(T)
\right\}.
\]
This refines the ordinary boundary-colouring relation by recording not
only which boundary colourings extend across $T$, but also how the
two-colour Kempe cycles are routed through the annular tire region.
\end{definition}
The annular structure of a tire is useful in two distinct ways. First,
it gives a bounded transition region between level cycles: the colouring
of the annular medial cycle controls, and in many cases determines, the
colouring of the remaining medial tire vertices. Thus the number of
possible transition states is bounded in terms of the annular structure,
rather than the total size of the subtree below the tire. Second, it
describes how the outer level cycle and the inner level cycles are
related by Kempe arcs. The level cycles are the gluing interfaces, while
the annular tire is the transition operator between them.
\begin{definition}[Kempe-compatible gluing]
Let $T$ be a medial tire region and let $U$ be a child region glued to
$T$ along a common level cycle $C$. Thus $C$ is an inner level cycle of
$T$ and the outer level cycle of $U$.
Let
\[
\kappa_T=(\alpha_T,\pi^T_{12},\pi^T_{23},\pi^T_{31})
\in \mathcal K_T
\]
and
\[
\kappa_U=(\alpha_U,\pi^U_{12},\pi^U_{23},\pi^U_{31})
\in \mathcal K_U.
\]
We say that $\kappa_T$ and $\kappa_U$ are \emph{Kempe-compatible along
$C$} if:
\begin{enumerate}
\item the boundary colourings agree on $C$:
\[
\alpha_T|_{V(C)}=\alpha_U|_{V(C)};
\]
\item for each colour pair
\[
P\in\{\{1,2\},\{2,3\},\{3,1\}\},
\]
the pairings $\pi^T_P$ and $\pi^U_P$ compose along the
$P$-coloured incidences of $C$ without producing an unpaired endpoint.
\end{enumerate}
When these conditions hold, the composed pairings determine a
Kempe-enhanced boundary state on the exposed boundary of
$T\cup_C U$.
\end{definition}
In these terms, gluing local colourings is not merely a matter of
matching boundary colours. The colourings must also route their
two-colour Kempe arcs compatibly across every shared level cycle. The
ordinary restriction relation records whether a boundary colouring can
be extended locally; the Kempe-enhanced relation additionally records
the conservation of Kempe-cycle flow through the annular transition
region.
For a tire with one outer level cycle and several inner level cycles,
\[
\partial T=C_0\sqcup C_1\sqcup\cdots\sqcup C_m,
\]
the parent tire may correlate the boundary states on the different
inner cycles. The Kempe-enhanced relation records this correlation as
a system of pairings among the $P$-coloured incidences of all boundary
level cycles simultaneously. Thus one should view a medial tire as a
multi-output transition operator
\[
\mathcal K_T:
C_0 \leadsto (C_1,\ldots,C_m),
\]
rather than as an independent collection of binary transitions.
The guiding principle is therefore:
\begin{quote}
Level cycles are the interfaces used for gluing, while annular tire
regions are the bounded transition regions that route Kempe cycles
between those interfaces.
\end{quote}
\begin{thebibliography}{9}
\bibitem{bauerfeld-nested-tire-decompositions}
@@ -666,11 +997,6 @@ E.~Bauerfeld,
\emph{Nested Tire Decompositions of Plane Triangulations},
manuscript (math-research repository), 2026.
\bibitem{bauerfeld-medial-pigeonhole}
E.~Bauerfeld,
\emph{The Medial Pigeonhole Programme},
manuscript (math-research repository), 2026.
\bibitem{tait-original}
P.~G. Tait,
\emph{Remarks on the colourings of maps},