From 20fe6c24ca82f7e07da5c59b751bf72dd228ffb3 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 8 Jun 2026 15:34:53 -0400 Subject: [PATCH] Add medial tire decomposition paper --- README.md | 1 + .../compare_full_reduced_medial_tires.py | 350 ++++++++++++ .../full_vs_reduced_medial_tire_findings.md | 82 +++ .../paper.aux | 34 ++ .../paper.fdb_latexmk | 138 +++++ .../paper.fls | 474 ++++++++++++++++ .../paper.log | 532 ++++++++++++++++++ .../paper.pdf | Bin 0 -> 232700 bytes .../paper.tex | 405 +++++++++++++ .../paper.aux | 43 +- .../paper.fdb_latexmk | 6 +- .../paper.log | 90 +-- .../paper.pdf | Bin 1082268 -> 1079047 bytes .../paper.tex | 51 +- 14 files changed, 2087 insertions(+), 119 deletions(-) create mode 100644 papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py create mode 100644 papers/medial_tire_decompositions_of_plane_triangulations/experiments/full_vs_reduced_medial_tire_findings.md create mode 100644 papers/medial_tire_decompositions_of_plane_triangulations/paper.aux create mode 100644 papers/medial_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk create mode 100644 papers/medial_tire_decompositions_of_plane_triangulations/paper.fls create mode 100644 papers/medial_tire_decompositions_of_plane_triangulations/paper.log create mode 100644 papers/medial_tire_decompositions_of_plane_triangulations/paper.pdf create mode 100644 papers/medial_tire_decompositions_of_plane_triangulations/paper.tex diff --git a/README.md b/README.md index f2733c7..06bda80 100644 --- a/README.md +++ b/README.md @@ -16,6 +16,7 @@ All papers are at `papers//paper.tex`. The current set: | `iterated_reduction_in_reduced_dual` | An Iterated Reduction in the Reduced Dual | | `level_resolutions_of_maximal_planar_graphs` | Level Resolutions of Maximal Planar Graphs | | `level_switching` | Level Switching | +| `medial_tire_decompositions_of_plane_triangulations` | Medial Tire Decompositions of Plane Triangulations | | `nested_tire_decompositions_of_plane_triangulations` | Nested Tire Decompositions of Plane Triangulations | | `plane_depth` | Plane Depth | | `plane_depth_sequencing` | Plane Depth Sequencing | diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py new file mode 100644 index 0000000..a10a6a9 --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py @@ -0,0 +1,350 @@ +"""Compare full and reduced medial tire graphs on generated tires. + +The new medial decomposition paper defines: + + * full medial tire graph: the subgraph of M(G) induced by medial + vertices corresponding to edges incident to tread triangles; + * reduced medial tire graph: delete same-boundary medial edges and + chord-only medial edges. + +For a tire tread inside an ambient triangulation, the medial edges +visible in the tread come from annular triangular faces. This script +checks whether any same-boundary medial edges are actually present in +that model. It also compares against the older standalone drawing +model, which added artificial outer/inner boundary faces. +""" + +from __future__ import annotations + +import argparse +import itertools +import random +from collections import Counter + + +Edge = tuple[int, int] +MedialEdge = tuple[Edge, Edge] + + +def random_tire(m: int, k: int, n_chords: int = 0, seed: int | None = None) -> dict: + """Generate the same labelled annular tires used in earlier experiments.""" + rng = random.Random(seed) + outer = list(range(m)) + inner = list(range(m, m + k)) + edges: set[Edge] = set() + + for i in range(m): + edges.add(edge_key(outer[i], outer[(i + 1) % m])) + for j in range(k): + edges.add(edge_key(inner[j], inner[(j + 1) % k])) + + inner_chords = set() + candidates = [] + for a in range(k): + for b in range(a + 2, k): + if not (a == 0 and b == k - 1): + candidates.append((a, b)) + rng.shuffle(candidates) + for a, b in candidates: + if len(inner_chords) >= n_chords: + break + if any((a < a2 < b < b2) or (a2 < a < b2 < b) for a2, b2 in inner_chords): + continue + inner_chords.add((a, b)) + edges.add(edge_key(inner[a], inner[b])) + + edges.add(edge_key(outer[0], inner[0])) + moves = ["O"] * m + ["I"] * k + rng.shuffle(moves) + triangles = [] + i, j = 0, 0 + for move in moves: + if move == "O": + tri = (outer[i % m], inner[j % k], outer[(i + 1) % m]) + triangles.append(tri) + edges.add(edge_key(inner[j % k], outer[(i + 1) % m])) + i += 1 + else: + tri = (outer[i % m], inner[j % k], inner[(j + 1) % k]) + triangles.append(tri) + edges.add(edge_key(outer[i % m], inner[(j + 1) % k])) + j += 1 + + return { + "m": m, + "k": k, + "n_chords": len(inner_chords), + "outer": outer, + "inner": inner, + "edges": sorted(edges), + "triangles": triangles, + "inner_chords": sorted(inner_chords), + "lattice_path": "".join(moves), + "seed": seed, + } + + +def tire_from_path(m: int, k: int, chords: tuple[tuple[int, int], ...], path: str) -> dict: + outer = list(range(m)) + inner = list(range(m, m + k)) + edges: set[Edge] = set() + + for i in range(m): + edges.add(edge_key(outer[i], outer[(i + 1) % m])) + for j in range(k): + edges.add(edge_key(inner[j], inner[(j + 1) % k])) + for a, b in chords: + edges.add(edge_key(inner[a], inner[b])) + + edges.add(edge_key(outer[0], inner[0])) + triangles = [] + i, j = 0, 0 + for move in path: + if move == "O": + tri = (outer[i % m], inner[j % k], outer[(i + 1) % m]) + triangles.append(tri) + edges.add(edge_key(inner[j % k], outer[(i + 1) % m])) + i += 1 + else: + tri = (outer[i % m], inner[j % k], inner[(j + 1) % k]) + triangles.append(tri) + edges.add(edge_key(outer[i % m], inner[(j + 1) % k])) + j += 1 + + return { + "m": m, + "k": k, + "n_chords": len(chords), + "outer": outer, + "inner": inner, + "edges": sorted(edges), + "triangles": triangles, + "inner_chords": sorted(chords), + "lattice_path": path, + "seed": None, + } + + +def chord_crosses(c1: tuple[int, int], c2: tuple[int, int]) -> bool: + a, b = c1 + c, d = c2 + return (a < c < b < d) or (c < a < d < b) + + +def chord_sets(k: int, max_chords: int) -> list[tuple[tuple[int, int], ...]]: + candidates = [] + for a in range(k): + for b in range(a + 2, k): + if not (a == 0 and b == k - 1): + candidates.append((a, b)) + + out = [()] + + def rec(start: int, chosen: tuple[tuple[int, int], ...]) -> None: + if len(chosen) >= max_chords: + return + for idx in range(start, len(candidates)): + chord = candidates[idx] + if any(chord_crosses(chord, old) for old in chosen): + continue + nxt = chosen + (chord,) + out.append(nxt) + rec(idx + 1, nxt) + + rec(0, ()) + return out + + +def lattice_paths(m: int, k: int): + for o_positions in itertools.combinations(range(m + k), m): + o_set = set(o_positions) + yield "".join("O" if idx in o_set else "I" for idx in range(m + k)) + + +def edge_key(u: int, v: int) -> Edge: + return tuple(sorted((u, v))) + + +def face_edges(face: tuple[int, ...]) -> list[Edge]: + return [edge_key(face[i], face[(i + 1) % len(face)]) for i in range(len(face))] + + +def is_cycle_edge(edge: Edge, cycle: list[int]) -> bool: + cycle_set = set(cycle) + if not set(edge) <= cycle_set: + return False + n = len(cycle) + idx = {v: i for i, v in enumerate(cycle)} + a, b = idx[edge[0]], idx[edge[1]] + return (a - b) % n in (1, n - 1) + + +def is_inner_chord(edge: Edge, m: int, k: int) -> bool: + u, v = edge + if not (m <= u < m + k and m <= v < m + k): + return False + a, b = u - m, v - m + d = abs(a - b) + return min(d, k - d) != 1 + + +def suppress_reason(e1: Edge, e2: Edge, tire: dict) -> str | None: + outer = tire["outer"] + inner = tire["inner"] + if is_cycle_edge(e1, outer) and is_cycle_edge(e2, outer): + return "outer_boundary" + if is_cycle_edge(e1, inner) and is_cycle_edge(e2, inner): + return "inner_boundary" + m, k = tire["m"], tire["k"] + if is_inner_chord(e1, m, k) or is_inner_chord(e2, m, k): + return "inner_chord" + return None + + +def medial_from_faces(faces: list[tuple[int, ...]], retained: set[Edge]) -> set[MedialEdge]: + medial_edges: set[MedialEdge] = set() + for face in faces: + boundary = [e for e in face_edges(face) if e in retained] + if len(boundary) < 2: + continue + for i, e in enumerate(boundary): + nxt = boundary[(i + 1) % len(boundary)] + if e != nxt: + medial_edges.add(tuple(sorted((e, nxt)))) + return medial_edges + + +def compare_tire(tire: dict, *, standalone_boundary_faces: bool) -> dict: + annular_faces = [tuple(tri) for tri in tire["triangles"]] + faces = list(annular_faces) + if standalone_boundary_faces: + faces.append(tuple(tire["outer"])) + faces.append(tuple(reversed(tire["inner"]))) + + # Definition 3.1 includes edges incident to at least one tread triangle. + retained = {e for face in annular_faces for e in face_edges(face)} + full_edges = medial_from_faces(faces, retained) + removed = {me for me in full_edges if suppress_reason(me[0], me[1], tire)} + reduced_edges = full_edges - removed + reasons = Counter(suppress_reason(me[0], me[1], tire) for me in removed) + reasons.pop(None, None) + return { + "vertices": len(retained), + "full_edges": len(full_edges), + "reduced_edges": len(reduced_edges), + "removed": len(removed), + "reasons": reasons, + "examples": sorted(removed)[:5], + } + + +def run_sweep(args: argparse.Namespace) -> None: + ambient_cases = 0 + ambient_differ = [] + standalone_cases = 0 + standalone_differ = [] + ambient_reasons: Counter[str] = Counter() + standalone_reasons: Counter[str] = Counter() + + max_chords = args.max_chords + for m in range(args.min_cycle, args.max_cycle + 1): + for k in range(args.min_cycle, args.max_cycle + 1): + for chords in range(max_chords + 1): + for seed in range(args.seeds): + tire = random_tire(m=m, k=k, n_chords=chords, seed=seed) + + ambient = compare_tire(tire, standalone_boundary_faces=False) + ambient_cases += 1 + ambient_reasons.update(ambient["reasons"]) + if ambient["removed"]: + ambient_differ.append((m, k, chords, seed, tire, ambient)) + + standalone = compare_tire(tire, standalone_boundary_faces=True) + standalone_cases += 1 + standalone_reasons.update(standalone["reasons"]) + if standalone["removed"]: + standalone_differ.append((m, k, chords, seed, tire, standalone)) + + print("ambient tread-face model") + print(f" cases checked: {ambient_cases}") + print(f" cases where full != reduced: {len(ambient_differ)}") + print(f" removed-edge reasons: {dict(sorted(ambient_reasons.items()))}") + if ambient_differ: + m, k, chords, seed, tire, result = ambient_differ[0] + print(" first difference:") + print(f" m={m} k={k} requested_chords={chords} seed={seed}") + print(f" path={tire['lattice_path']} chords={tire['inner_chords']}") + print(f" removed examples={result['examples']}") + + print() + print("standalone tire-with-boundary-faces model") + print(f" cases checked: {standalone_cases}") + print(f" cases where full != reduced: {len(standalone_differ)}") + print(f" removed-edge reasons: {dict(sorted(standalone_reasons.items()))}") + if standalone_differ: + m, k, chords, seed, tire, result = standalone_differ[0] + print(" first difference:") + print(f" m={m} k={k} requested_chords={chords} seed={seed}") + print(f" path={tire['lattice_path']} chords={tire['inner_chords']}") + print(f" full_edges={result['full_edges']} reduced_edges={result['reduced_edges']}") + print(f" removed examples={result['examples']}") + + +def run_exhaustive(args: argparse.Namespace) -> None: + ambient_cases = 0 + ambient_differ = [] + standalone_cases = 0 + standalone_differ = [] + for m in range(args.min_cycle, args.max_cycle + 1): + for k in range(args.min_cycle, args.max_cycle + 1): + for chords in chord_sets(k, args.max_chords): + for path in lattice_paths(m, k): + tire = tire_from_path(m, k, chords, path) + + ambient = compare_tire(tire, standalone_boundary_faces=False) + ambient_cases += 1 + if ambient["removed"]: + ambient_differ.append((m, k, chords, path, ambient)) + + standalone = compare_tire(tire, standalone_boundary_faces=True) + standalone_cases += 1 + if standalone["removed"]: + standalone_differ.append((m, k, chords, path, standalone)) + + print("exhaustive ambient tread-face model") + print(f" cases checked: {ambient_cases}") + print(f" cases where full != reduced: {len(ambient_differ)}") + if ambient_differ: + m, k, chords, path, result = ambient_differ[0] + print(" first difference:") + print(f" m={m} k={k} chords={chords} path={path}") + print(f" removed examples={result['examples']}") + + print() + print("exhaustive standalone tire-with-boundary-faces model") + print(f" cases checked: {standalone_cases}") + print(f" cases where full != reduced: {len(standalone_differ)}") + if standalone_differ: + m, k, chords, path, result = standalone_differ[0] + print(" first difference:") + print(f" m={m} k={k} chords={chords} path={path}") + print(f" full_edges={result['full_edges']} reduced_edges={result['reduced_edges']}") + print(f" removed examples={result['examples']}") + + +def main() -> None: + parser = argparse.ArgumentParser() + parser.add_argument("--min-cycle", type=int, default=3) + parser.add_argument("--max-cycle", type=int, default=8) + parser.add_argument("--max-chords", type=int, default=3) + parser.add_argument("--seeds", type=int, default=50) + parser.add_argument("--exhaustive", action="store_true") + args = parser.parse_args() + if args.exhaustive: + run_exhaustive(args) + else: + run_sweep(args) + + +if __name__ == "__main__": + main() diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/full_vs_reduced_medial_tire_findings.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/full_vs_reduced_medial_tire_findings.md new file mode 100644 index 0000000..e0b8e42 --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/full_vs_reduced_medial_tire_findings.md @@ -0,0 +1,82 @@ +# Full vs Reduced Medial Tire Findings + +Question: do Definition 3.1 (full medial tire graph) and Definition 3.2 +(reduced medial tire graph) differ? + +## Experiment + +Script: + +```bash +python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py +``` + +The script compares two models. + +- Ambient tread-face model: medial edges are contributed by annular + triangular faces of the tire tread inside the ambient triangulation. +- Standalone tire-with-boundary-faces model: the outer and inner + boundary walks are also treated as faces, as in the older drawing + script. + +## Random Sweep + +Command: + +```bash +python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py +``` + +Result: + +```text +ambient tread-face model + cases checked: 7200 + cases where full != reduced: 0 + removed-edge reasons: {} + +standalone tire-with-boundary-faces model + cases checked: 7200 + cases where full != reduced: 7200 + removed-edge reasons: {'inner_boundary': 39600, 'outer_boundary': 39600} + first difference: + m=3 k=3 requested_chords=0 seed=0 + path=IOOOII chords=[] + full_edges=24 reduced_edges=18 + removed examples=[((0, 1), (0, 2)), ((0, 1), (1, 2)), ((0, 2), (1, 2)), ((3, 4), (3, 5)), ((3, 4), (4, 5))] +``` + +## Exhaustive Small Sweep + +Command: + +```bash +python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py --exhaustive --max-cycle 5 --max-chords 2 +``` + +Result: + +```text +exhaustive ambient tread-face model + cases checked: 5578 + cases where full != reduced: 0 + +exhaustive standalone tire-with-boundary-faces model + cases checked: 5578 + cases where full != reduced: 5578 + first difference: + m=3 k=3 chords=() path=OOOIII + full_edges=24 reduced_edges=18 + removed examples=[((0, 1), (0, 2)), ((0, 1), (1, 2)), ((0, 2), (1, 2)), ((3, 4), (3, 5)), ((3, 4), (4, 5))] +``` + +## Interpretation + +For the intended ambient-triangulation definition, the experiments +support the suspicion that Definition 3.1 and Definition 3.2 coincide: +same-boundary medial edges do not arise from annular triangular tread +faces, and inner chords are not incident to tread triangles. + +They differ only in the standalone tire-with-boundary-faces model, +where the artificial outer and inner boundary faces create medial edges +between consecutive boundary edges. diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/paper.aux b/papers/medial_tire_decompositions_of_plane_triangulations/paper.aux new file mode 100644 index 0000000..3dc986c --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/paper.aux @@ -0,0 +1,34 @@ +\relax +\citation{bauerfeld-nested-tire-decompositions} +\citation{bauerfeld-nested-tire-decompositions} +\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Background}}{1}{}\protected@file@percent } +\citation{bauerfeld-nested-tire-decompositions} +\citation{bauerfeld-nested-tire-decompositions} +\newlabel{def:medial-graph}{{2.1}{2}} +\newlabel{prop:medial-dual-invariance}{{2.3}{2}} +\newlabel{cor:tait-medial}{{2.4}{2}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Medial tire pieces}}{2}{}\protected@file@percent } +\newlabel{def:full-medial-tire}{{3.1}{2}} +\newlabel{thm:annular-medial-colour-bound}{{3.3}{3}} +\newlabel{def:boundary-medial-vertices}{{3.4}{3}} +\newlabel{def:medial-restriction-relation}{{3.5}{3}} +\citation{bauerfeld-nested-tire-decompositions} +\citation{bauerfeld-nested-tire-decompositions} +\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Decomposition}}{4}{}\protected@file@percent } +\newlabel{cor:medial-tire-decomposition}{{4.1}{4}} +\newlabel{def:compatible-family}{{4.2}{4}} +\newlabel{prop:gluing-criterion}{{4.3}{4}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{A medial pigeonhole programme}}{4}{}\protected@file@percent } +\bibcite{bauerfeld-nested-tire-decompositions}{1} +\bibcite{tait-original}{2} +\newlabel{tocindent-1}{0pt} +\newlabel{tocindent0}{12.7778pt} +\newlabel{tocindent1}{17.77782pt} +\newlabel{tocindent2}{0pt} +\newlabel{tocindent3}{0pt} +\newlabel{def:medial-boundary-state}{{5.1}{5}} +\newlabel{conj:medial-chain-pigeonhole}{{5.2}{5}} +\newlabel{conj:medial-route-fct}{{5.3}{5}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{5}{}\protected@file@percent } +\gdef \@abspage@last{5} diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk b/papers/medial_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk new file mode 100644 index 0000000..c9286f3 --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk @@ -0,0 +1,138 @@ +# Fdb version 3 +["pdflatex"] 1780947173 "paper.tex" "paper.pdf" "paper" 1780947174 + "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 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+\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{Medial Tire Decompositions of Plane Triangulations} + +% 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} +We use the nested tire decomposition of a plane triangulation to induce +a decomposition of its full medial graph into medial tire subgraphs. +For a plane triangulation $G$, the medial graph $M(G)$ is naturally +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, +their boundary-state restriction relations, and a chain-pigeonhole +conjecture for compatible medial boundary states across the tire tree. +\end{abstract} + +\maketitle + +\section{Introduction} + +A classical theorem of Tait recasts the Four Colour Theorem in dual, +edge-colouring terms: a plane triangulation $G$ is properly +$4$-vertex-colourable if and only if its dual cubic graph $G^*$ is +properly $3$-edge-colourable. The present paper records a medial +version of this viewpoint. The vertices of the medial graph $M(G)$ +correspond to edges of $G$, and adjacency in $M(G)$ records +consecutiveness of edges around vertices and faces of $G$. Since +planar duality interchanges vertices and faces while preserving the +edge set, $M(G)$ is naturally isomorphic to $M(G^*)$. + +Consequently a proper vertex $3$-colouring of $M(G)$ is the same +object as a proper edge $3$-colouring of $G^*$. This suggests another +route toward the Four Colour Theorem: rather than colouring the dual +cubic graph directly, decompose the full medial graph into local +annular pieces and try to prove that their proper vertex +$3$-colouring boundary restrictions always compose. + +The structural input is the nested tire decomposition of +\cite{bauerfeld-nested-tire-decompositions}. A level source in a plane +triangulation determines a rooted tree of tire treads. Each tread is +an annular triangulated region with an outer boundary, an inner +outerplanar graph, and annular triangular faces. We show that this +decomposition induces a decomposition of $M(G)$ into medial tire +subgraphs. The boundary data of a medial tire are proper +$3$-colourings of the medial vertices corresponding to boundary edges +in the associated dual tire graph. + +\section{Background} + +Throughout, $G$ is a simple plane maximal planar graph with fixed +embedding, and $G^*$ denotes its full planar dual. We use the level +source, dual depth, tire graph, tire tread, and tire-tree terminology +of~\cite{bauerfeld-nested-tire-decompositions}. In particular, a level +source $S$ determines a rooted tire tree $\mathcal{T}(G,S)$ whose +vertices are tire treads and whose parent-child relation records +nested containment across level-cycle interfaces. + +\begin{definition}[Medial graph] +\label{def:medial-graph} +Let $H$ be a plane graph. The \emph{medial graph} $M(H)$ has one +vertex $m_e$ for each edge $e \in E(H)$. Two medial vertices +$m_e,m_f$ are adjacent whenever $e$ and $f$ are consecutive in the +cyclic order of edges around a vertex of $H$ or around a face of $H$. +The embedding is the standard one obtained by placing $m_e$ at the +midpoint of $e$ and drawing medial edges through the vertex- and +face-corners of $H$. +\end{definition} + +\begin{remark} +If $H$ has bridges or vertices of degree $1$, the usual medial +construction may create parallel edges or loops depending on the +chosen convention. In this paper the main application is to plane +triangulations and their cubic planar duals, where the medial graph is +a loopless $4$-regular plane graph. +\end{remark} + +\begin{proposition}[Medial dual invariance] +\label{prop:medial-dual-invariance} +Let $H$ be a connected plane graph and let $H^*$ be its planar dual. +Then there is a natural plane-graph isomorphism +\[ + M(H) \cong M(H^*). +\] +\end{proposition} + +\begin{proof} +Each edge $e \in E(H)$ corresponds to a unique dual edge $e^* \in +E(H^*)$, giving a bijection $m_e \mapsto m_{e^*}$ between the vertices +of $M(H)$ and $M(H^*)$. In $M(H)$ two vertices $m_e,m_f$ are adjacent +exactly when $e$ and $f$ are consecutive around either a vertex or a +face of $H$. Under duality, vertices and faces are interchanged, and +the cyclic order of the corresponding dual edges around the dual face +or dual vertex is the same up to reversal. Thus the same pairs are +medial-adjacent in $M(H^*)$, and the midpoint construction identifies +the two embedded medial graphs. +\end{proof} + +\begin{corollary}[Tait colourings as medial vertex colourings] +\label{cor:tait-medial} +Let $G$ be a simple plane triangulation. Proper vertex +$3$-colourings of $M(G)$ are in natural bijection with proper +$3$-edge-colourings of the cubic planar dual $G^*$. +\end{corollary} + +\begin{proof} +By Proposition~\ref{prop:medial-dual-invariance}, $M(G) \cong +M(G^*)$. Vertices of $M(G^*)$ correspond to edges of $G^*$, and two +such vertices are adjacent exactly when the corresponding dual edges +are incident and consecutive around a vertex or face of $G^*$. Since +$G^*$ is cubic, proper vertex $3$-colouring of $M(G^*)$ is therefore +equivalent to assigning three colours to the edges of $G^*$ so that the +three edges incident to each dual vertex receive pairwise distinct +colours. +\end{proof} + +\section{Medial tire pieces} + +\begin{definition}[Full medial tire graph] +\label{def:full-medial-tire} +Let $T$ be a tire tread in the tire tree $\mathcal{T}(G,S)$ supplied +by~\cite{bauerfeld-nested-tire-decompositions}. The \emph{full medial +tire graph} of $T$, denoted $\mathsf{M}(T)$, is the subgraph of +$M(G)$ induced by the medial vertices $m_e$ with $e$ an edge of $G$ +incident to at least one triangular face in the tread $T$. The medial +vertices corresponding to annular edges of $T$ are called +\emph{annular medial vertices}. +\end{definition} + +\begin{remark} +In the ambient-triangulation setting, the full medial tire graph +$\mathsf{M}(T)$ coincides with the omitted-edge medial tire graph +studied in~\cite{bauerfeld-nested-tire-decompositions}. Indeed, the +medial edges of $\mathsf{M}(T)$ are contributed by corners of annular +triangular tread faces. Such a face contains at most one outer-boundary +edge and at most one inner-boundary edge, so it does not contribute a +medial edge between two outer-boundary edges or between two +inner-boundary edges. Similarly, chords of the inner outerplanar graph +lie outside the annular tread and are not incident to annular tread +faces. Thus the deletion rule used for the earlier reduced medial tire +graph removes no edges from the ambient object $\mathsf{M}(T)$. + +The distinction only appears in the standalone drawing convention where +the outer and inner boundary walks are added as artificial faces before +forming a medial graph. Those artificial faces create same-boundary +medial edges, and the reduced construction deletes them. +\end{remark} + +\begin{theorem}[Annular medial colour bound] +\label{thm:annular-medial-colour-bound} +Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire tread with +non-degenerate boundaries and simple inner boundary $B_{\mathrm{in}}$. +Let $A(T)$ be the subgraph of $\mathsf{M}(T)$ induced by the annular +medial vertices. For a graph $H$, write $\operatorname{Col}_3(H)$ for +the set of proper $3$-vertex-colourings of $H$. Then $A(T)$ is a cycle +and +\[ + |\operatorname{Col}_3(\mathsf{M}(T))| + \;\leq\; |\operatorname{Col}_3(A(T))|. +\] +\end{theorem} + +\begin{proof} +Since the tread is a triangulated annulus with no vertices in its +interior, each annular face has exactly one boundary edge, lying either +on $B_{\mathrm{out}}$ or on $B_{\mathrm{in}}$, and exactly two annular +edges. As the annular faces are traversed cyclically around the tread, +consecutive faces share one annular edge. Equivalently, the annular +edges occur in a cyclic order in which each annular face contains two +consecutive annular edges. Hence the subgraph of $\mathsf{M}(T)$ +induced by the annular medial vertices is a cycle. + +Consider the restriction map from proper $3$-colourings of +$\mathsf{M}(T)$ to colourings of this annular medial cycle $A(T)$. We +claim that this map is injective. Let $x$ be a non-annular medial +vertex. Then $x$ corresponds to an edge of $B_{\mathrm{out}}$ or +$B_{\mathrm{in}}$: chords of $O$ are not incident to annular tread +faces, and hence do not contribute vertices of $\mathsf{M}(T)$. This +boundary edge is incident to a unique annular face of the tread, and +the other two edges of that face are annular edges. Therefore $x$ is +adjacent in $\mathsf{M}(T)$ to the two annular medial vertices +corresponding to those two annular edges. + +Those two annular medial vertices are adjacent to each other, because +their annular edges are consecutive on the same triangular annular +face. In any proper $3$-colouring they therefore receive two distinct +colours, and $x$ is forced to receive the remaining third colour. Thus +every non-annular medial vertex has its colour uniquely determined by +the colouring of $A(T)$. Two colourings of $\mathsf{M}(T)$ with the +same restriction to $A(T)$ are identical, so the restriction map is +injective. The stated inequality follows. +\end{proof} + +\begin{definition}[Boundary medial vertices] +\label{def:boundary-medial-vertices} +Let $T$ be a tire tread and let $\Gamma_T$ be the corresponding dual +tire subgraph in $G^*$. A vertex $m_e \in V(\mathsf{M}(T))$ is an +\emph{outer boundary medial vertex} if the corresponding dual edge +$e^* \in E(G^*)$ lies on the outer boundary of $\Gamma_T$. It is an +\emph{inner boundary medial vertex} if $e^*$ lies on the inner boundary +of $\Gamma_T$. We write +\[ + \partial_{\mathrm{out}}\mathsf{M}(T) + \quad\text{and}\quad + \partial_{\mathrm{in}}\mathsf{M}(T) +\] +for the two boundary sets. +\end{definition} + +\begin{definition}[Medial tire restriction relation] +\label{def:medial-restriction-relation} +Let $\mathrm{Col}_3(X)$ denote the set of proper vertex +$3$-colourings of the induced subgraph on a vertex set $X$. The +\emph{medial tire restriction relation} of $T$ is +\[ + R_T \subseteq + \mathrm{Col}_3(\partial_{\mathrm{out}}\mathsf{M}(T)) + \times + \mathrm{Col}_3(\partial_{\mathrm{in}}\mathsf{M}(T)), +\] +where $(\alpha,\beta) \in R_T$ exactly when $\alpha \cup \beta$ +extends to a proper vertex $3$-colouring of $\mathsf{M}(T)$. +\end{definition} + +\begin{remark} +The definition deliberately records boundary colourings on medial +vertices corresponding to boundary edges in the dual tire graph. Under +Corollary~\ref{cor:tait-medial}, these are precisely edge-colouring +states on the boundary edges through which a dual tire piece meets its +parent and children. +\end{remark} + +\section{Decomposition} + +\begin{corollary}[Medial tire decomposition] +\label{cor:medial-tire-decomposition} +Let $G$ be a plane triangulation with level source $S$. The tire-tree +decomposition $\mathcal{T}(G,S)$ of +\cite{bauerfeld-nested-tire-decompositions} induces a rooted +decomposition of the full medial graph $M(G)$ into full medial tire +graphs $\{\mathsf{M}(T): T \in V(\mathcal{T}(G,S))\}$, glued along +their boundary medial vertex sets. +\end{corollary} + +\begin{proof} +By the tire-tread partition theorem of +\cite{bauerfeld-nested-tire-decompositions}, the bounded triangular +faces of $G$ are partitioned into nested tire treads, with intersections +between parent and child treads occurring only along their level-cycle +interface data. Every edge of $G$ that is incident to a bounded face +therefore belongs to the closure of at least one tire tread, and an +edge lying in two closures lies on the interface between adjacent +treads in the tire tree. Passing to $M(G)$ sends edges of $G$ to +medial vertices. Thus each tread determines the induced subgraph +$\mathsf{M}(T)$ on its incident edge set, and overlaps between two such +subgraphs are exactly the medial vertices corresponding to interface +edges, namely the appropriate boundary medial vertex sets. +\end{proof} + +\begin{definition}[Compatible family of medial tire colourings] +\label{def:compatible-family} +A \emph{compatible family of medial tire colourings} on +$\mathcal{T}(G,S)$ is a choice, for each tread $T$, of a proper +vertex $3$-colouring $\varphi_T$ of $\mathsf{M}(T)$ such that whenever +$T'$ is a child tread of $T$, the two colourings agree on +$ + V(\mathsf{M}(T)) \cap V(\mathsf{M}(T')). +$ +\end{definition} + +\begin{proposition}[Gluing criterion] +\label{prop:gluing-criterion} +The full medial graph $M(G)$ has a proper vertex $3$-colouring if and +only if the tire tree $\mathcal{T}(G,S)$ admits a compatible family of +medial tire colourings. +\end{proposition} + +\begin{proof} +A proper vertex $3$-colouring of $M(G)$ restricts to a proper vertex +$3$-colouring of every induced subgraph $\mathsf{M}(T)$, and these +restrictions agree on overlaps. + +Conversely, suppose a compatible family is given. Define a colour on +each vertex $m_e$ of $M(G)$ by choosing any tread $T$ with +$m_e \in V(\mathsf{M}(T))$ and setting +$\varphi(m_e)=\varphi_T(m_e)$. Compatibility makes this independent of +the choice of $T$. Every medial edge of $M(G)$ is drawn in a corner of +some bounded triangular face of $G$ or along the outer boundary +interface. The relevant incident primal edges lie together in the +closure of a single tire tread or in a shared boundary interface, where +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} + +\begin{thebibliography}{9} + +\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} diff --git a/papers/nested_tire_decompositions_of_plane_triangulations/paper.aux b/papers/nested_tire_decompositions_of_plane_triangulations/paper.aux index 9caf7f0..72701f1 100644 --- a/papers/nested_tire_decompositions_of_plane_triangulations/paper.aux +++ b/papers/nested_tire_decompositions_of_plane_triangulations/paper.aux @@ -8,50 +8,48 @@ \citation{dvorak-lidicky-cones} \citation{heesch-untersuchungen} \citation{robertson-sanders-seymour-thomas} +\citation{robertson-sanders-seymour-thomas} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Related work.}}{1}{}\protected@file@percent } -\citation{robertson-sanders-seymour-thomas} \newlabel{def:dual}{{1.3}{2}} \newlabel{def:dual-depth}{{1.4}{2}} \newlabel{def:dual-component}{{1.5}{2}} +\newlabel{def:tire-graph}{{1.6}{2}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{3}{}\protected@file@percent } \newlabel{fig:dual-depth}{{1}{3}} -\newlabel{def:tire-graph}{{1.6}{3}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{4}{}\protected@file@percent } \newlabel{fig:tire-example}{{2}{4}} \newlabel{def:medial-tire-graph}{{1.7}{4}} -\newlabel{thm:annular-medial-colour-bound}{{1.8}{4}} +\newlabel{rem:tire-counts}{{1.8}{4}} +\newlabel{prop:no-level-d-pinch}{{1.9}{4}} \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The medial tire graph for the tire in Figure\nonbreakingspace 2\hbox {}. The chord of $O$ is drawn faintly and omitted before taking the medial graph; medial edges between consecutive outer-boundary edges or consecutive inner-boundary edges are also omitted. Each medial vertex is placed at the midpoint of its corresponding retained tire edge.}}{5}{}\protected@file@percent } \newlabel{fig:medial-tire-example}{{3}{5}} -\newlabel{rem:tire-counts}{{1.9}{5}} -\newlabel{prop:no-level-d-pinch}{{1.10}{6}} -\newlabel{lem:tire-component}{{1.11}{6}} \citation{bauerfeld-depth} +\newlabel{lem:tire-component}{{1.10}{6}} \citation{bauerfeld-depth} -\newlabel{thm:tread-partition}{{1.12}{8}} -\newlabel{rem:tire-component-degenerate}{{1.13}{8}} -\newlabel{rem:tire-no-extra-hypotheses}{{1.14}{8}} -\newlabel{thm:inner-dual-outerplanar}{{1.15}{9}} -\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Case 1 ($R$ = disk, $k = 6$). The apex $v_0$ sits at the centre; the non-degenerate boundary $B_{\mathrm {non-deg}}$ (red) is the hexagonal outer cycle; spokes (grey) triangulate the disk into a fan of $6$ triangles around $v_0$. Each triangle has two spoke edges (interior, contributing $\Gamma $-edges) and one boundary edge (contributing a leaf in $D(T)$, no $\Gamma $-edge). The inner dual $\Gamma $ (blue) is the cycle $C_6$ formed by the six annular face centroids, a manifestly outerplanar graph.}}{10}{}\protected@file@percent } -\newlabel{fig:inner-dual-disk-case}{{4}{10}} +\newlabel{thm:tread-partition}{{1.11}{7}} +\newlabel{rem:tire-component-degenerate}{{1.12}{8}} +\newlabel{rem:tire-no-extra-hypotheses}{{1.13}{8}} +\newlabel{thm:inner-dual-outerplanar}{{1.14}{8}} +\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Case 1 ($R$ = disk, $k = 6$). The apex $v_0$ sits at the centre; the non-degenerate boundary $B_{\mathrm {non-deg}}$ (red) is the hexagonal outer cycle; spokes (grey) triangulate the disk into a fan of $6$ triangles around $v_0$. Each triangle has two spoke edges (interior, contributing $\Gamma $-edges) and one boundary edge (contributing a leaf in $D(T)$, no $\Gamma $-edge). The inner dual $\Gamma $ (blue) is the cycle $C_6$ formed by the six annular face centroids, a manifestly outerplanar graph.}}{9}{}\protected@file@percent } +\newlabel{fig:inner-dual-disk-case}{{4}{9}} \citation{bauerfeld-nested-tire-duals} \citation{bauerfeld-nested-tire-duals} +\newlabel{rem:hamilton-cycle-spoke-only}{{1.15}{10}} +\newlabel{rem:bridge-case-theta}{{1.16}{10}} +\newlabel{thm:tread-tree}{{1.17}{10}} \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Case 2 ($R$ = annulus) with $O$ a barbell. $B_{\mathrm {out}}$ is the outer hexagon (red); $O$ has two triangles $\{a_1, a_2, a_3\}$ and $\{b_1, b_2, b_3\}$ joined by the bridge $a_3\text {--}b_1$ (all light red). The annulus is triangulated by $14$ annular triangles: $6$ ``outer-cap'' triangles (one per outer edge), $6$ ``inner-cap'' triangles (one per non-bridge edge of $O$), and $2$ ``bridge-cap'' triangles $\{u_0, a_3, b_1\}$ and $\{u_3, a_3, b_1\}$ adjacent to the bridge. Each blue dot sits at the centroid of an annular triangle; blue edges connect dual vertices whose triangles share an interior annular edge (spoke or bridge). The two bridge-cap vertices have $\Gamma $-degree $3$ (their triangles have no boundary edge) and are joined by the dashed blue \emph {chord} corresponding to the bridge; the remaining $13$ edges form the Hamilton cycle that wraps around the annulus. All $14$ vertices lie on the outer face of the cycle-with-chord embedding, so $\Gamma \cong \Theta (1, 7, 7)$ is outerplanar.}}{11}{}\protected@file@percent } \newlabel{fig:inner-dual-annulus-case}{{5}{11}} -\newlabel{rem:hamilton-cycle-spoke-only}{{1.16}{11}} -\newlabel{rem:bridge-case-theta}{{1.17}{11}} -\newlabel{thm:tread-tree}{{1.18}{12}} -\newlabel{rem:tree-multiple-children}{{1.19}{13}} -\newlabel{thm:tire-tree-decomposition}{{1.20}{13}} -\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.20\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{15}{}\protected@file@percent } -\newlabel{fig:tire-tree-decomposition}{{6}{15}} -\newlabel{rem:tree-coloring-factorisation}{{1.21}{15}} +\newlabel{rem:tree-multiple-children}{{1.18}{12}} +\newlabel{thm:tire-tree-decomposition}{{1.19}{12}} \bibcite{tait-original}{1} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tire-duals}{3} \bibcite{birkhoff-reducibility}{4} \bibcite{birkhoff-lewis-chromatic}{5} \bibcite{tutte-four-colour-conjecture}{6} +\newlabel{rem:tree-coloring-factorisation}{{1.20}{14}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{14}{}\protected@file@percent } \bibcite{tutte-algebraic-colorings}{7} \bibcite{tutte-chromatic-sums-1973}{8} \bibcite{heesch-untersuchungen}{9} @@ -62,5 +60,6 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{16}{}\protected@file@percent } -\gdef \@abspage@last{16} +\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.19\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{15}{}\protected@file@percent } +\newlabel{fig:tire-tree-decomposition}{{6}{15}} +\gdef \@abspage@last{15} diff --git a/papers/nested_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk b/papers/nested_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk index d9afbbb..5c02c47 100644 --- a/papers/nested_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk +++ b/papers/nested_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1780945550 "paper.tex" "paper.pdf" "paper" 1780945552 +["pdflatex"] 1780947176 "paper.tex" "paper.pdf" "paper" 1780947178 "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -143,8 +143,8 @@ "fig_medial_tire_example.png" 1780943640 209003 4349824e4f016bde4b938be6b2cb5b2c "" "fig_tire_example.png" 1779857443 104494 8f9ce26b469b4236b8b67829f73a5faa "" "fig_tire_tree_decomposition.png" 1780290287 372371 1b44f5a3e9f637d78ae951b1f2e3a89d "" - "paper.aux" 1780945552 6952 b88a9a8b764572f2fc3be94aabc8b157 "pdflatex" - "paper.tex" 1780945491 60644 1d96ad32143c562e73fccdfacbbc9be2 "" + "paper.aux" 1780947178 6896 03aab2e2511678ab091cd0356763e615 "pdflatex" + "paper.tex" 1780947122 58249 66f46a4d7e93e153ac47781a196e8b3d "" (generated) "paper.aux" "paper.log" diff --git a/papers/nested_tire_decompositions_of_plane_triangulations/paper.log b/papers/nested_tire_decompositions_of_plane_triangulations/paper.log index 3948fb5..8d317de 100644 --- a/papers/nested_tire_decompositions_of_plane_triangulations/paper.log +++ b/papers/nested_tire_decompositions_of_plane_triangulations/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 8 JUN 2026 15:05 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 8 JUN 2026 15:32 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -498,7 +498,7 @@ e File: fig_dual_depth.png Graphic file (type png) -Package pdftex.def Info: fig_dual_depth.png used on input line 163. +Package pdftex.def Info: fig_dual_depth.png used on input line 162. 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z^Q`_keleuoU$(&V?Pns@D;Qz=C+ZUOqJ*?7{K+^sPtpq69 zMCLg~tt8H(u-{Z?GAz2vT=AGIGINyTUpQw`oIRQT&gZ>>V=c1QVMmLjshCy0r4z}hG)s`FsORt6iKf1Yx4tpA+oWO;cmP-WY9 z_P1ol`dif#8epf^jb(jz{D6!$BnD@tMQAOf`iW9IhuD-@+o>#atEi8<@5%yT%?~1T zqENq46H&+jIjV3Cg0m7woJ5EjI#L)UGn6hF7eOUf2J`xJMUHlC>$8ut^IHXYEQCwu zHP&(hm$smiiOtqB%ZifpfU5E*4p1lu$Z5=Ymj)Z6KD*7+LGK+lhR~K7wBZ3DM&kL# P!O4L@MJ1^$h46m>y?CWp diff --git a/papers/nested_tire_decompositions_of_plane_triangulations/paper.tex b/papers/nested_tire_decompositions_of_plane_triangulations/paper.tex index d84f606..4b1296a 100644 --- a/papers/nested_tire_decompositions_of_plane_triangulations/paper.tex +++ b/papers/nested_tire_decompositions_of_plane_triangulations/paper.tex @@ -52,9 +52,8 @@ $G$ induces a BFS layering of $G$ and endows the inner planar dual $G'$ with a \emph{dual depth} grading. The basic object of study is the \emph{tire graph} $T$ --- a plane graph whose outer and inner boundaries bound a closed planar region, the \emph{tire tread} $R$, -triangulated by the \emph{annular edges} $E_{\mathrm{ann}}$. We define -medial tire graphs and prove a basic colour-count bound for their -annular medial cycle. Our main structural results are the +triangulated by the \emph{annular edges} $E_{\mathrm{ann}}$. Our main +structural results are the \emph{tire-component lemma}, the \emph{tire-tread partition theorem}, and the rooted \emph{tire-tree decomposition}, which together organize the bounded faces of $G$ into nested tire treads. @@ -283,52 +282,6 @@ corresponding retained tire edge.} \label{fig:medial-tire-example} \end{figure} -\begin{theorem}[Annular medial colour bound] -\label{thm:annular-medial-colour-bound} -Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph with -non-degenerate boundaries and simple inner boundary $B_{\mathrm{in}}$. -Let $A(T)$ be the subgraph of $M_{\mathrm{tire}}(T)$ induced by the -annular medial vertices. For a graph $H$, write -$\operatorname{Col}_3(H)$ for the set of proper $3$-vertex-colourings -of $H$. Then $A(T)$ is a cycle and -\[ - |\operatorname{Col}_3(M_{\mathrm{tire}}(T))| - \;\leq\; |\operatorname{Col}_3(A(T))|. -\] -\end{theorem} - -\begin{proof} -Since the tread is a triangulated annulus with no vertices in its -interior, each annular face has exactly one boundary edge, lying either -on $B_{\mathrm{out}}$ or on $B_{\mathrm{in}}$, and exactly two annular -edges. As the annular faces are traversed cyclically around the tread, -consecutive faces share one annular edge. Equivalently, the annular -edges occur in a cyclic order in which each annular face contains two -consecutive annular edges. Hence the subgraph of -$M_{\mathrm{tire}}(T)$ induced by the annular medial vertices is a -cycle. - -Consider the restriction map from proper $3$-colourings of -$M_{\mathrm{tire}}(T)$ to colourings of this annular medial cycle -$A(T)$. We claim that this map is injective. Let $x$ be a -non-annular medial vertex. Then $x$ corresponds to an edge of -$B_{\mathrm{out}}$ or $B_{\mathrm{in}}$, since the chords of $O$ were -omitted before forming $M_{\mathrm{tire}}(T)$. This boundary edge is -incident to a unique annular face of $T^{\circ}$, and the other two -edges of that face are annular edges. Therefore $x$ is adjacent in -$M_{\mathrm{tire}}(T)$ to the two annular medial vertices corresponding -to those two annular edges. - -Those two annular medial vertices are adjacent to each other, because -their annular edges are consecutive on the same triangular annular -face. In any proper $3$-colouring they therefore receive two distinct -colours, and $x$ is forced to receive the remaining third colour. -Thus every non-annular medial vertex has its colour uniquely determined -by the colouring of $A(T)$. Two colourings of $M_{\mathrm{tire}}(T)$ -with the same restriction to $A(T)$ are identical, so the restriction -map is injective. The stated inequality follows. -\end{proof} - \begin{remark} \label{rem:tire-counts} Let $\mu = |V(B_{\mathrm{out}})|$ and $\nu = |V(B_{\mathrm{in}})|$. By