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papers: split coloring_nested_tire foundations into separate paper

Browse Source 
NEW PAPER: papers/coloring_nested_tire_graphs/ ("Coloring Nested
Tire Graphs", 5 pages).

Contains foundational definitions 1.1 through 1.7 from the dual
paper, plus the four illustrative figures:
  - 1.1 Level source
  - 1.2 Levels
  - 1.3 Dual (with label def:dual added — was missing in original)
  - 1.4 Dual depth
  - 1.5 Tire graph
  - 1.6 Remark (tire counts)
  - 1.7 Partial tire dual

Also: the dual-depth figure, the tire-example figure, and both
partial-tire-dual figures (vanilla + bridge case).

MODIFIED: papers/coloring_nested_tire_dual_graphs/paper.tex now a
follow-up:
  - Abstract recasts the paper as building on the foundational paper.
  - Intro no longer recapitulates definitions; lists them as
    citations to the new paper.
  - Removes definitions 1.1-1.7 and their figures (now in
    foundational paper).
  - Internal \ref{...} to removed labels converted to
    \cite[Definition N.M]{bauerfeld-nested-tires}.
  - Bibliography adds the new paper as a reference.
  - Renumbering: theorems/propositions now start at 1.1 (formerly
    1.8). Paper down from 14 to 8 pages.

Both papers compile cleanly with no broken references.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-27 00:54:53 -04:00
parent c234e0d2dd
commit 65f79f2e65
12 changed files with 687 additions and 304 deletions
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papers/coloring_nested_tire_dual_graphs/paper.aux
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\relax \relax
\citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires}
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\@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.}}{2}{}\protected@file@percent } \citation{bauerfeld-nested-tires}
\newlabel{fig:dual-depth}{{1}{2}} \newlabel{prop:no-level-d-pinch}{{1.2}{2}}
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\@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.}}{3}{}\protected@file@percent }
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\newlabel{lem:tire-component}{{1.10}{6}} \citation{bauerfeld-nested-tires}
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\citation{bauerfeld-depth} \citation{bauerfeld-depth}
\newlabel{rem:tire-component-degenerate}{{1.11}{8}} \citation{bauerfeld-nested-tires}
\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{8}} \newlabel{rem:tire-component-degenerate}{{1.4}{4}}
\newlabel{prop:edge-vertex-bijection}{{1.13}{8}} \newlabel{rem:tire-no-extra-hypotheses}{{1.5}{4}}
\newlabel{rem:edge-vertex-corollary}{{1.14}{9}} \newlabel{prop:edge-vertex-bijection}{{1.6}{4}}
\newlabel{def:tire-annular-subgraph}{{1.15}{9}} \citation{bauerfeld-nested-tires}
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\newlabel{def:spokes}{{1.17}{9}} \newlabel{rem:edge-vertex-corollary}{{1.7}{5}}
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\newlabel{conj:chain-latin}{{2.2}{7}}
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\OMS/cmsy/m/n/10 L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 B[]; O\OT1/cmr/m/n/10 ) := \OMS/cmsy/m/n/10 L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 B[]; O\OT1/cmr/m/n/10 ) :=
[] \OML/cmm/m/it/10 ^^[ \OT1/cmr/m/n/10 : \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 ( [] \OML/cmm/m/it/10 ^^[ \OT1/cmr/m/n/10 : \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (
\OML/cmm/m/it/10 B[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ! f\OT1/cmr/m/n/10 1\OML \OML/cmm/m/it/10 B[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ! f\OT1/cmr/m/n/10 1\OML
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\OML/cmm/m/it/10 O\OT1/cmr/m/n/10 ) []\OML/cmm/m/it/10 : \OML/cmm/m/it/10 O\OT1/cmr/m/n/10 ) []\OML/cmm/m/it/10 :
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papers/coloring_nested_tire_dual_graphs/paper.pdf
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papers/coloring_nested_tire_dual_graphs/paper.tex
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\dedicatory{} \dedicatory{}
\begin{abstract} \begin{abstract}
% TODO: abstract. This is a follow-up to \cite{bauerfeld-nested-tires}, which
establishes the basic vocabulary of tire graphs $T$ and their
partial tire duals $D(T)$. Building on those definitions, we
analyse the structure of $D(T)$ in the spoke-only case (a corona
graph $C_{n+m} \circ K_1$), prove the tire-component lemma
exhibiting every BFS-level component as a tire graph, give an
edge-vertex coloring bijection that reduces counting proper
$3$-edge-colorings of $D(T)$ to counting proper $3$-vertex-colorings
of a cycle, and develop the tire-annular-subgraph, face-connector,
and inner/outer-spoke structures in $G'$. A concluding section
records a Latin-substructure conjecture for chain-pigeonhole
compatibility of adjacent tires.
\end{abstract} \end{abstract}
\maketitle \maketitle
@@ -58,205 +69,33 @@ minimal counterexample to the Four Colour Theorem -- a smallest triangulation
admitting no proper $4$-colouring -- corresponds to a smallest cubic plane graph admitting no proper $4$-colouring -- corresponds to a smallest cubic plane graph
admitting no proper $3$-edge-colouring. admitting no proper $3$-edge-colouring.
We study the structure such a minimal counterexample would have to exhibit This paper is the second in a series studying that structure
through the lens of \emph{nested level duals}. Fixing a level source $S$ in $G$ through the lens of \emph{nested level duals}. The foundational
endows the dual $G'$ with a Breadth-First-Search--derived labelling, the dual vocabulary --- level sources, levels, the inner planar dual $G'$
depth of Definition~\ref{def:dual-depth}, and the level structure of $G$ organises and its dual depth, tire graphs, and partial tire duals
$G'$ into a family of nested cycles carrying these labels. Our aim is to express $D(T)$ --- is developed in the companion paper
the obstruction to a $3$-edge-colouring of $G'$ as conditions on this nested \cite{bauerfeld-nested-tires}; we refer to that paper for all
labelled-cycle structure. basic definitions and rely on them throughout. In particular we
use, without restating, the notions of:
\begin{itemize}
\item \emph{level source} $S$ and $G$-vertex levels $\ell_G(v)$;
\item the inner planar dual $G'$
(\cite[Definition~1.3]{bauerfeld-nested-tires});
\item \emph{dual depth} $\delta_G(d_f)$
(\cite[Definition~1.4]{bauerfeld-nested-tires});
\item \emph{tire graph} $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$
with outer/inner boundaries and annular edges
(\cite[Definition~1.5]{bauerfeld-nested-tires});
\item \emph{partial tire dual} $D(T)$
(\cite[Definition~1.7]{bauerfeld-nested-tires});
\item face/edge counts
(\cite[Remark~1.6]{bauerfeld-nested-tires}).
\end{itemize}
Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation) Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation)
with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$ with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$
and $G$ has $2n - 4$ triangular faces. and $G$ has $2n - 4$ triangular faces.
\begin{definition}[Level source]
A \emph{level source} of $G$ is any vertex $v \in V$; we write
$S = \{v\}$ for the level-0 source.
\end{definition}
\begin{definition}[Levels]
Given a level source $S \subseteq V$, the \emph{level} of $v \in V$ is
$\ell_G(v) = \mathrm{dist}_G(v, S)$, the graph distance from $v$ to the nearest
source vertex.
\end{definition}
\begin{definition}[Dual]
The \emph{dual} of $G$, written $G'$, is the inner (weak) planar dual of $G$ with
respect to the embedding $\Pi_G$: it has one vertex $d_f$ for each bounded face
$f$ of $G$, and an edge joining $d_f$ and $d_{f'}$ for each edge of $G$ shared by
two bounded faces $f$ and $f'$. The unbounded outer face contributes no vertex,
and edges of $G$ on the outer boundary contribute no dual edge. Since $G$ is a
triangulation, each vertex $d_f \in V(G')$ corresponds to a triangular face $f$
of $G$, and we write $V(f) \subseteq V$ for its three incident vertices.
\end{definition}
\begin{definition}[Dual depth]
\label{def:dual-depth}
Given a level source $S \subseteq V$, the \emph{dual depth} of a dual vertex
$d_f \in V(G')$ is
\[
\delta_G(d_f) = \min_{v \in V(f)} \ell_G(v)
= \min_{v \in V(f)} \mathrm{dist}_G(v, S),
\]
the smallest level among the three vertices of $G$ bounding the face $f$.
\end{definition}
\begin{figure}[h]
\centering
\includegraphics[width=0.7\textwidth]{fig_dual_depth.png}
\caption{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 $\delta(d_f) = \min_{v \in V(f)} \ell(v)$: 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.}
\label{fig:dual-depth}
\end{figure}
\begin{definition}[Tire graph]
\label{def:tire-graph}
A \emph{tire graph} consists of a plane graph $T$ together with an
\emph{outer boundary} $B_{\mathrm{out}} \subseteq T$ and an \emph{inner
outerplanar graph} $O \subseteq T$ with $V(B_{\mathrm{out}}) \cap V(O)
= \emptyset$, where
\begin{itemize}
\item $B_{\mathrm{out}}$ is either a simple cycle of length $\geq 3$
or a single vertex (a \emph{degenerate outer boundary});
\item $O$ is an outerplanar graph; its \emph{inner boundary}
$B_{\mathrm{in}}$ is the closed walk in $O$ that traces the
boundary of $O$'s outer face in the inherited embedding,
which is a simple cycle when $O$ is $2$-connected and a
non-simple closed walk in general (visiting bridges twice and
cut-vertices multiple times); if $|V(O)| = 1$, we say $T$ has
a \emph{degenerate inner boundary}.
\end{itemize}
At most one of $B_{\mathrm{out}}, B_{\mathrm{in}}$ may be degenerate.
The vertex and edge sets of $T$ are
\[
V(T) = V(B_{\mathrm{out}}) \cup V(O),
\qquad
E(T) = E(B_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}},
\]
where $E_{\mathrm{ann}}$ --- the \emph{annular edges} --- has the
property that, in the plane embedding of $T$, the closed planar region
$R$ bounded externally by $B_{\mathrm{out}}$ and internally by
$B_{\mathrm{in}}$ is partitioned into triangular faces of $T$ whose
union is $R$.
When $B_{\mathrm{out}}$ is a simple cycle and $O$ is $2$-connected,
$R$ is a closed annulus. More generally, $R$ is a closed planar
region that may fail to be a $2$-manifold at cut-vertices of $O$ (where
two ``lobes'' of the depth-$d$ region meet at a single vertex); the
inner boundary $B_{\mathrm{in}}$ is then a non-simple closed walk that
visits the cut-vertex multiple times. The relaxed definition
accommodates outerplanar inner graphs with bridges, cut-vertices, or
multiple connected components. When either boundary is degenerate,
$R$ is a closed disk with that vertex as apex.
\end{definition}
\begin{figure}[h]
\centering
\includegraphics[width=0.78\textwidth]{fig_tire_example.png}
\caption{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.}
\label{fig:tire-example}
\end{figure}
\begin{remark}
\label{rem:tire-counts}
Let $m = |V(B_{\mathrm{out}})|$ and $k = |V(B_{\mathrm{in}})|$. By
Euler's formula on the annular (resp.\ disk) region $R$, the tire graph
has $m + k$ triangular faces inside $R$ and $|E_{\mathrm{ann}}| = m + k$
annular edges when neither boundary is degenerate; when exactly one
boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$
triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$.
\end{remark}
\begin{definition}[Partial tire dual]
\label{def:partial-tire-dual}
Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph in
the sense of Definition~\ref{def:tire-graph}, and let $F_{\mathrm{ann}}$
denote the set of triangular faces of $T$ in the closed annular region
between $B_{\mathrm{out}}$ and $B_{\mathrm{in}}$. The \emph{partial
tire dual} of $T$, written $D(T)$, is the graph defined as follows.
\emph{Vertices.}
\begin{enumerate}
\item[(V1)] For each face $f \in F_{\mathrm{ann}}$, an
\emph{interior vertex} $d_f$ of $D(T)$.
\item[(V2)] For each edge $e \in E(B_{\mathrm{out}})$, a
\emph{leaf vertex} $\ell_e^{\mathrm{out}}$.
\item[(V3)] For each occurrence of an edge in the closed walk
$B_{\mathrm{in}}$ (= the outer-face boundary walk of $O$),
a \emph{leaf vertex} $\ell_e^{\mathrm{in}}$. (When $O$ is
$2$-connected each edge appears once; cut-vertices and
bridges of $O$ may cause an edge or vertex to appear more
than once.)
\end{enumerate}
\emph{Edges.}
\begin{enumerate}
\item[(E1)] For each edge $e \in E(T)$ whose two incident faces both
lie in $F_{\mathrm{ann}}$ (an \emph{interior annular edge}),
one edge $\{d_{f_1}, d_{f_2}\} \in E(D(T))$ where
$f_1, f_2 \in F_{\mathrm{ann}}$ are the two annular faces
incident to $e$.
\item[(E2)] For each $e \in E(B_{\mathrm{out}})$, one edge
$\{d_f, \ell_e^{\mathrm{out}}\} \in E(D(T))$ where
$f \in F_{\mathrm{ann}}$ is the unique annular face incident
to $e$. The leaf $\ell_e^{\mathrm{out}}$ has degree $1$.
\item[(E3)] For each occurrence of $e$ on the boundary walk
$B_{\mathrm{in}}$, one edge
$\{d_f, \ell_e^{\mathrm{in}}\} \in E(D(T))$ where
$f \in F_{\mathrm{ann}}$ is the annular face incident to $e$
on the side of that occurrence. The leaf
$\ell_e^{\mathrm{in}}$ has degree $1$.
\end{enumerate}
\end{definition}
\begin{figure}[h]
\centering
\includegraphics[width=0.78\textwidth]{fig_partial_tire_dual.png}
\caption{The partial tire dual $D(T)$ (purple squares + orange
diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$
and $k = 4$. The ten interior vertices $d_f$ at the centroids of the
annular triangles form a single $10$-cycle (solid purple); each
boundary edge of the annular region (either of $B_{\mathrm{out}}$ or
of $B_{\mathrm{in}}$) contributes a degree-$1$ leaf (orange diamond)
attached to the unique annular face incident to it (dashed orange),
giving the structure $C_{10} \circ K_1$ of
Proposition~\ref{prop:partial-tire-dual-structure}.}
\label{fig:partial-tire-dual-example}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.85\textwidth]{fig_partial_tire_dual_bridge.png}
\caption{Partial tire dual $D(T)$ when the inner outerplanar graph
$O$ has a bridge --- here a non-trivial edge cut connecting two
disjoint triangles. $B_{\mathrm{out}}$ is a $4$-cycle on
$\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together
with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing
the bridge disconnects $O$). Because both faces incident to the
bridge are annular triangles, the bridge contributes an
\emph{interior dual edge} (highlighted in red) rather than two
leaves; consequently the interior dual subgraph is no longer the
single $(n+m)$-cycle of
Proposition~\ref{prop:partial-tire-dual-structure}, but a theta
graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident
annular faces) are joined by three internally vertex-disjoint paths
in $D(T)$. Leaves come only from $B_{\mathrm{out}}$ ($n = 4$ leaves)
and the six non-bridge edges of $O$ ($m_{\partial} = 6$ leaves,
three for each triangle).}
\label{fig:partial-tire-dual-bridge}
\end{figure}
\begin{proposition}[Structure of $D(T)$ when the annular triangulation \begin{proposition}[Structure of $D(T)$ when the annular triangulation
is spoke-only] is spoke-only]
@@ -288,7 +127,7 @@ So each $d_f$ has degree $3$ in $D(T)$: two from interior edges (= spokes
shared with adjacent annular faces) and one leaf. The induced subgraph shared with adjacent annular faces) and one leaf. The induced subgraph
on $\{d_f : f \in F_{\mathrm{ann}}\}$ is $2$-regular; together with the on $\{d_f : f \in F_{\mathrm{ann}}\}$ is $2$-regular; together with the
connectedness of the annular region this forces it to be a single connectedness of the annular region this forces it to be a single
cycle. By Remark~\ref{rem:tire-counts}, the cycle has length $n + m$, cycle. By \cite[Remark~1.6]{bauerfeld-nested-tires}, the cycle has length $n + m$,
and there are also $n + m$ leaves attached one-per-cycle-vertex. and there are also $n + m$ leaves attached one-per-cycle-vertex.
\end{proof} \end{proof}
@@ -371,7 +210,7 @@ its embedding from $\Pi_G$. Set $R_{C'} := \bigcup_{f \in F_{C'}} f
\subseteq |\Pi_G|$. \subseteq |\Pi_G|$.
Then $C$, with the inherited embedding, is a tire graph in the sense of Then $C$, with the inherited embedding, is a tire graph in the sense of
Definition~\ref{def:tire-graph}. Its outer boundary \cite[Definition~1.5]{bauerfeld-nested-tires}. Its outer boundary
$B_{\mathrm{out}}$ is the side of $R_{C'}$ closer to $S$ in $\Pi_G$, $B_{\mathrm{out}}$ is the side of $R_{C'}$ closer to $S$ in $\Pi_G$,
namely the level-$d$ subgraph $G[V_{C'} \cap L_d]$ (a simple cycle or namely the level-$d$ subgraph $G[V_{C'} \cap L_d]$ (a simple cycle or
single vertex); its inner outerplanar graph is $O = G[V_{C'} \cap single vertex); its inner outerplanar graph is $O = G[V_{C'} \cap
@@ -434,7 +273,7 @@ cycle. At vertices $v \in L_{d+1} \cap V_{C'}$ the depth-$d$ faces
may split into multiple arcs of $v$'s rotation; this corresponds may split into multiple arcs of $v$'s rotation; this corresponds
exactly to $v$ being a cut-vertex of $O$, and the inner-side exactly to $v$ being a cut-vertex of $O$, and the inner-side
boundary walk visits $v$ correspondingly many times --- which is boundary walk visits $v$ correspondingly many times --- which is
already accommodated by Definition~\ref{def:tire-graph} (where already accommodated by \cite[Definition~1.5]{bauerfeld-nested-tires} (where
$B_{\mathrm{in}}$ is the outer-face boundary closed walk of $O$, not $B_{\mathrm{in}}$ is the outer-face boundary closed walk of $O$, not
necessarily a simple cycle). necessarily a simple cycle).
@@ -487,7 +326,7 @@ the level-$D_{\max}$ cycle as the outer boundary.
\label{rem:tire-no-extra-hypotheses} \label{rem:tire-no-extra-hypotheses}
Two structural features of $R_{C'}$ that might at first appear to Two structural features of $R_{C'}$ that might at first appear to
obstruct the tire-graph conclusion are both already accommodated by obstruct the tire-graph conclusion are both already accommodated by
Definition~\ref{def:tire-graph}: \cite[Definition~1.5]{bauerfeld-nested-tires}:
\emph{Cut-vertices of $O$.} A vertex $v \in V_{C'} \cap L_{d+1}$ may \emph{Cut-vertices of $O$.} A vertex $v \in V_{C'} \cap L_{d+1}$ may
have the faces of $F_{C'}$ incident to it split into two or more have the faces of $F_{C'}$ incident to it split into two or more
@@ -574,9 +413,9 @@ its attached interior vertex.
\begin{definition}[Tire annular subgraph] \begin{definition}[Tire annular subgraph]
\label{def:tire-annular-subgraph} \label{def:tire-annular-subgraph}
Let $G$ be a maximal planar graph with embedding $\Pi_G$ and inner Let $G$ be a maximal planar graph with embedding $\Pi_G$ and inner
planar dual $G'$ (as in Definition~\ref{def:dual} above). Let planar dual $G'$ (as in \cite[Definition~1.3]{bauerfeld-nested-tires} above). Let
$T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}}) \subseteq G$ be a tire $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}}) \subseteq G$ be a tire
graph (Definition~\ref{def:tire-graph}), and let graph (\cite[Definition~1.5]{bauerfeld-nested-tires}), and let
$F_{\mathrm{ann}} \subseteq F(G)$ denote its set of annular faces. $F_{\mathrm{ann}} \subseteq F(G)$ denote its set of annular faces.
The \emph{tire annular subgraph} of $T$ in $G'$ is The \emph{tire annular subgraph} of $T$ in $G'$ is
\[ \[
@@ -760,6 +599,11 @@ E.~Bauerfeld,
\emph{Plane Depth}, \emph{Plane Depth},
manuscript (math-research repository), 2026. manuscript (math-research repository), 2026.
\bibitem{bauerfeld-nested-tires}
E.~Bauerfeld,
\emph{Coloring Nested Tire Graphs},
manuscript (math-research repository), 2026.
\end{thebibliography} \end{thebibliography}
\end{document} \end{document}
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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}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
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\newtheorem{xca}[theorem]{Exercise}
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\begin{document}
\title{Coloring Nested Tire Graphs}
% author one information
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{}
\thanks{}
\subjclass[2010]{Primary }
\keywords{plane graph, triangulation, plane depth, level edge, dual graph, tire graph}
\date{}
\dedicatory{}
\begin{abstract}
We establish the foundational definitions for studying the
Four Colour Theorem through nested level-structures on plane
triangulations. A \emph{level source} of a triangulation $G$
induces a BFS layering of $G$, which in turn endows the inner
planar dual $G'$ with a \emph{dual depth} grading. We isolate the
basic object of study --- the \emph{tire graph} $T$, a plane graph
whose outer and inner boundaries bound an annular region
triangulated by the \emph{annular edges} $E_{\mathrm{ann}}$ --- and
define its \emph{partial tire dual} $D(T)$, the dual restricted to
$T$'s annular faces together with leaves recording the boundary
edges.
\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. Thus a
minimal counterexample to the Four Colour Theorem -- a smallest triangulation
admitting no proper $4$-colouring -- corresponds to a smallest cubic plane graph
admitting no proper $3$-edge-colouring.
The structural study of such a minimal counterexample is the
overarching motivation for the present line of work. This first
paper establishes the foundational vocabulary --- level sources,
dual depth, tire graphs, and partial tire duals --- on which
subsequent papers in the series build. In particular, the
companion paper \cite{bauerfeld-nested-tire-duals} uses these
definitions to develop nested-cycle structure theorems and
chain-pigeonhole conjectures for tire annular subgraphs of $G'$.
Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation)
with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$
and $G$ has $2n - 4$ triangular faces.
\begin{definition}[Level source]
A \emph{level source} of $G$ is any vertex $v \in V$; we write
$S = \{v\}$ for the level-0 source.
\end{definition}
\begin{definition}[Levels]
Given a level source $S \subseteq V$, the \emph{level} of $v \in V$ is
$\ell_G(v) = \mathrm{dist}_G(v, S)$, the graph distance from $v$ to the nearest
source vertex.
\end{definition}
\begin{definition}[Dual]
\label{def:dual}
The \emph{dual} of $G$, written $G'$, is the inner (weak) planar dual of $G$ with
respect to the embedding $\Pi_G$: it has one vertex $d_f$ for each bounded face
$f$ of $G$, and an edge joining $d_f$ and $d_{f'}$ for each edge of $G$ shared by
two bounded faces $f$ and $f'$. The unbounded outer face contributes no vertex,
and edges of $G$ on the outer boundary contribute no dual edge. Since $G$ is a
triangulation, each vertex $d_f \in V(G')$ corresponds to a triangular face $f$
of $G$, and we write $V(f) \subseteq V$ for its three incident vertices.
\end{definition}
\begin{definition}[Dual depth]
\label{def:dual-depth}
Given a level source $S \subseteq V$, the \emph{dual depth} of a dual vertex
$d_f \in V(G')$ is
\[
\delta_G(d_f) = \min_{v \in V(f)} \ell_G(v)
= \min_{v \in V(f)} \mathrm{dist}_G(v, S),
\]
the smallest level among the three vertices of $G$ bounding the face $f$.
\end{definition}
\begin{figure}[h]
\centering
\includegraphics[width=0.7\textwidth]{fig_dual_depth.png}
\caption{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 $\delta(d_f) = \min_{v \in V(f)} \ell(v)$: 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.}
\label{fig:dual-depth}
\end{figure}
\begin{definition}[Tire graph]
\label{def:tire-graph}
A \emph{tire graph} consists of a plane graph $T$ together with an
\emph{outer boundary} $B_{\mathrm{out}} \subseteq T$ and an \emph{inner
outerplanar graph} $O \subseteq T$ with $V(B_{\mathrm{out}}) \cap V(O)
= \emptyset$, where
\begin{itemize}
\item $B_{\mathrm{out}}$ is either a simple cycle of length $\geq 3$
or a single vertex (a \emph{degenerate outer boundary});
\item $O$ is an outerplanar graph; its \emph{inner boundary}
$B_{\mathrm{in}}$ is the closed walk in $O$ that traces the
boundary of $O$'s outer face in the inherited embedding,
which is a simple cycle when $O$ is $2$-connected and a
non-simple closed walk in general (visiting bridges twice and
cut-vertices multiple times); if $|V(O)| = 1$, we say $T$ has
a \emph{degenerate inner boundary}.
\end{itemize}
At most one of $B_{\mathrm{out}}, B_{\mathrm{in}}$ may be degenerate.
The vertex and edge sets of $T$ are
\[
V(T) = V(B_{\mathrm{out}}) \cup V(O),
\qquad
E(T) = E(B_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}},
\]
where $E_{\mathrm{ann}}$ --- the \emph{annular edges} --- has the
property that, in the plane embedding of $T$, the closed planar region
$R$ bounded externally by $B_{\mathrm{out}}$ and internally by
$B_{\mathrm{in}}$ is partitioned into triangular faces of $T$ whose
union is $R$.
When $B_{\mathrm{out}}$ is a simple cycle and $O$ is $2$-connected,
$R$ is a closed annulus. More generally, $R$ is a closed planar
region that may fail to be a $2$-manifold at cut-vertices of $O$ (where
two ``lobes'' of the depth-$d$ region meet at a single vertex); the
inner boundary $B_{\mathrm{in}}$ is then a non-simple closed walk that
visits the cut-vertex multiple times. The relaxed definition
accommodates outerplanar inner graphs with bridges, cut-vertices, or
multiple connected components. When either boundary is degenerate,
$R$ is a closed disk with that vertex as apex.
\end{definition}
\begin{figure}[h]
\centering
\includegraphics[width=0.78\textwidth]{fig_tire_example.png}
\caption{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.}
\label{fig:tire-example}
\end{figure}
\begin{remark}
\label{rem:tire-counts}
Let $m = |V(B_{\mathrm{out}})|$ and $k = |V(B_{\mathrm{in}})|$. By
Euler's formula on the annular (resp.\ disk) region $R$, the tire graph
has $m + k$ triangular faces inside $R$ and $|E_{\mathrm{ann}}| = m + k$
annular edges when neither boundary is degenerate; when exactly one
boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$
triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$.
\end{remark}
\begin{definition}[Partial tire dual]
\label{def:partial-tire-dual}
Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph in
the sense of Definition~\ref{def:tire-graph}, and let $F_{\mathrm{ann}}$
denote the set of triangular faces of $T$ in the closed annular region
between $B_{\mathrm{out}}$ and $B_{\mathrm{in}}$. The \emph{partial
tire dual} of $T$, written $D(T)$, is the graph defined as follows.
\emph{Vertices.}
\begin{enumerate}
\item[(V1)] For each face $f \in F_{\mathrm{ann}}$, an
\emph{interior vertex} $d_f$ of $D(T)$.
\item[(V2)] For each edge $e \in E(B_{\mathrm{out}})$, a
\emph{leaf vertex} $\ell_e^{\mathrm{out}}$.
\item[(V3)] For each occurrence of an edge in the closed walk
$B_{\mathrm{in}}$ (= the outer-face boundary walk of $O$),
a \emph{leaf vertex} $\ell_e^{\mathrm{in}}$. (When $O$ is
$2$-connected each edge appears once; cut-vertices and
bridges of $O$ may cause an edge or vertex to appear more
than once.)
\end{enumerate}
\emph{Edges.}
\begin{enumerate}
\item[(E1)] For each edge $e \in E(T)$ whose two incident faces both
lie in $F_{\mathrm{ann}}$ (an \emph{interior annular edge}),
one edge $\{d_{f_1}, d_{f_2}\} \in E(D(T))$ where
$f_1, f_2 \in F_{\mathrm{ann}}$ are the two annular faces
incident to $e$.
\item[(E2)] For each $e \in E(B_{\mathrm{out}})$, one edge
$\{d_f, \ell_e^{\mathrm{out}}\} \in E(D(T))$ where
$f \in F_{\mathrm{ann}}$ is the unique annular face incident
to $e$. The leaf $\ell_e^{\mathrm{out}}$ has degree $1$.
\item[(E3)] For each occurrence of $e$ on the boundary walk
$B_{\mathrm{in}}$, one edge
$\{d_f, \ell_e^{\mathrm{in}}\} \in E(D(T))$ where
$f \in F_{\mathrm{ann}}$ is the annular face incident to $e$
on the side of that occurrence. The leaf
$\ell_e^{\mathrm{in}}$ has degree $1$.
\end{enumerate}
\end{definition}
\begin{figure}[h]
\centering
\includegraphics[width=0.78\textwidth]{fig_partial_tire_dual.png}
\caption{The partial tire dual $D(T)$ (purple squares + orange
diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$
and $k = 4$. The ten interior vertices $d_f$ at the centroids of the
annular triangles form a single $10$-cycle (solid purple); each
boundary edge of the annular region (either of $B_{\mathrm{out}}$ or
of $B_{\mathrm{in}}$) contributes a degree-$1$ leaf (orange diamond)
attached to the unique annular face incident to it (dashed orange),
giving the structure $C_{10} \circ K_1$ analysed in the companion
paper~\cite{bauerfeld-nested-tire-duals}.}
\label{fig:partial-tire-dual-example}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.85\textwidth]{fig_partial_tire_dual_bridge.png}
\caption{Partial tire dual $D(T)$ when the inner outerplanar graph
$O$ has a bridge --- here a non-trivial edge cut connecting two
disjoint triangles. $B_{\mathrm{out}}$ is a $4$-cycle on
$\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together
with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing
the bridge disconnects $O$). Because both faces incident to the
bridge are annular triangles, the bridge contributes an
\emph{interior dual edge} (highlighted in red) rather than two
leaves; consequently the interior dual subgraph is no longer the
single $(n+m)$-cycle of the spoke-only case, but a theta
graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident
annular faces) are joined by three internally vertex-disjoint paths
in $D(T)$. Leaves come only from $B_{\mathrm{out}}$ ($n = 4$ leaves)
and the six non-bridge edges of $O$ ($m_{\partial} = 6$ leaves,
three for each triangle).}
\label{fig:partial-tire-dual-bridge}
\end{figure}
\begin{thebibliography}{9}
\bibitem{bauerfeld-depth}
E.~Bauerfeld,
\emph{Plane Depth},
manuscript (math-research repository), 2026.
\bibitem{bauerfeld-nested-tire-duals}
E.~Bauerfeld,
\emph{Coloring Nested Tire Dual Graphs},
manuscript (math-research repository), 2026.
\end{thebibliography}
\end{document}