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Add Medial Tire Cuts paper with walk-depth labelling and cut

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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>
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didericis
2026-06-14 21:39:03 -04:00
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commit 291f7e98c7
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\citation{bauerfeld-medial-tire}
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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}
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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{thebibliography}{9}
\bibitem{bauerfeld-medial-tire}
E.~Bauerfeld,
\emph{Medial Tire Decompositions of Plane Triangulations},
manuscript (math-research repository), 2026.
\end{thebibliography}
\end{document}