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coloring_nested_tire_graphs: notation cleanup pass

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Define previously-implicit objects and unify conventions:
- define level sets L_d (and L_{<d}, L_{>=d}) in the Levels definition
- factor G'_d, F_{C'}, V_{C'}, R_{C'} into a standalone definition
  before Prop 1.6, removing the forward reference
- name the annular faces F_ann and state the tire-graph tuple form
  T = (B_out, O, E_ann) in the tire-graph definition
- ground the full tire dual D(T) where Gamma is introduced
- normalize tree superscripts (0)/(p)/(c) to the tire-symbol form
  (T_0)/(T_p)/(T_c)
- resolve the boundary-count clash: use nu = |V(B_in)| (inner) and
  mu = |V(B_out)| (outer) throughout, freeing n for |V(G)|

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-29 23:38:12 -04:00
parent 454c79b289
commit 92f5efc3f1
4 changed files with 165 additions and 126 deletions
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papers/coloring_nested_tire_graphs/paper.aux
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\relax \relax
\citation{bauerfeld-nested-tire-duals} \citation{bauerfeld-nested-tire-duals}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
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\newlabel{def:dual-depth}{{1.4}{2}} \newlabel{def:dual-depth}{{1.4}{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.}}{2}{}\protected@file@percent } \@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 }
\newlabel{fig:dual-depth}{{1}{2}} \newlabel{fig:dual-depth}{{1}{2}}
\newlabel{def:tire-graph}{{1.5}{2}} \newlabel{def:dual-component}{{1.5}{2}}
\@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 } \newlabel{def:tire-graph}{{1.6}{3}}
\newlabel{fig:tire-example}{{2}{3}} \newlabel{rem:tire-counts}{{1.7}{3}}
\newlabel{rem:tire-counts}{{1.6}{3}} \newlabel{prop:no-level-d-pinch}{{1.8}{3}}
\newlabel{prop:no-level-d-pinch}{{1.7}{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{lem:tire-component}{{1.8}{4}} \newlabel{fig:tire-example}{{2}{4}}
\citation{bauerfeld-depth} \citation{bauerfeld-depth}
\newlabel{lem:tire-component}{{1.9}{5}}
\citation{bauerfeld-depth} \citation{bauerfeld-depth}
\newlabel{thm:tread-partition}{{1.9}{6}} \newlabel{thm:tread-partition}{{1.10}{6}}
\newlabel{rem:tire-component-degenerate}{{1.10}{6}} \newlabel{rem:tire-component-degenerate}{{1.11}{7}}
\newlabel{rem:tire-no-extra-hypotheses}{{1.11}{6}} \newlabel{rem:tire-no-extra-hypotheses}{{1.12}{7}}
\newlabel{thm:inner-dual-outerplanar}{{1.12}{7}} \newlabel{thm:inner-dual-outerplanar}{{1.13}{7}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\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.}}{8}{}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {3}{\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.}}{8}{}\protected@file@percent }
\newlabel{fig:inner-dual-disk-case}{{3}{8}} \newlabel{fig:inner-dual-disk-case}{{3}{8}}
\citation{bauerfeld-nested-tire-duals} \citation{bauerfeld-nested-tire-duals}
\citation{bauerfeld-nested-tire-duals} \citation{bauerfeld-nested-tire-duals}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\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.}}{9}{}\protected@file@percent }
\newlabel{fig:inner-dual-annulus-case}{{4}{9}}
\newlabel{rem:hamilton-cycle-spoke-only}{{1.13}{9}}
\newlabel{rem:bridge-case-theta}{{1.14}{9}}
\citation{tait-original} \citation{tait-original}
\newlabel{thm:tait-tire}{{1.15}{10}} \newlabel{rem:hamilton-cycle-spoke-only}{{1.14}{9}}
\newlabel{rem:count-general-outerplanar}{{1.16}{10}} \newlabel{rem:bridge-case-theta}{{1.15}{9}}
\newlabel{thm:tread-tree}{{1.17}{10}} \newlabel{thm:tait-tire}{{1.16}{9}}
\newlabel{rem:tree-multiple-children}{{1.18}{11}} \@writefile{lof}{\contentsline {figure}{\numberline {4}{\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.}}{10}{}\protected@file@percent }
\newlabel{thm:tire-tree-decomposition}{{1.19}{12}} \newlabel{fig:inner-dual-annulus-case}{{4}{10}}
\newlabel{rem:tree-coloring-factorisation}{{1.20}{13}} \newlabel{rem:count-general-outerplanar}{{1.17}{11}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\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 $(a)$.}}{14}{}\protected@file@percent } \newlabel{thm:tread-tree}{{1.18}{11}}
\newlabel{fig:tire-tree-decomposition}{{5}{14}} \newlabel{rem:tree-multiple-children}{{1.19}{12}}
\newlabel{def:seam}{{1.21}{14}} \newlabel{thm:tire-tree-decomposition}{{1.20}{12}}
\newlabel{def:partial-tire-tree}{{1.22}{15}} \newlabel{rem:tree-coloring-factorisation}{{1.21}{14}}
\newlabel{lem:seam-edge-shared}{{1.23}{15}} \newlabel{def:seam}{{1.22}{14}}
\newlabel{conj:seam-counterexample}{{1.24}{15}} \newlabel{def:partial-tire-tree}{{1.23}{14}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\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}{{5}{15}}
\newlabel{lem:seam-edge-shared}{{1.24}{15}}
\bibcite{tait-original}{1} \bibcite{tait-original}{1}
\bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-depth}{2}
\bibcite{bauerfeld-nested-tire-duals}{3} \bibcite{bauerfeld-nested-tire-duals}{3}
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@@ -98,7 +98,10 @@ A \emph{level source} of $G$ is a set $S \subseteq V$ that is either
\begin{definition}[Levels] \begin{definition}[Levels]
Given a level source $S \subseteq V$, the \emph{level} of $v \in V$ is 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 $\ell_G(v) = \mathrm{dist}_G(v, S)$, the graph distance from $v$ to the nearest
source vertex. source vertex. We write $L_d := \{v \in V : \ell_G(v) = d\}$ for the
\emph{level-$d$ vertex set}, and abbreviate $L_{<d} := \bigcup_{d' < d}
L_{d'}$ and $L_{\geq d} := \bigcup_{d' \geq d} L_{d'}$ (similarly
$L_{>d}$, $L_{\leq d}$).
\end{definition} \end{definition}
\begin{definition}[Dual] \begin{definition}[Dual]
@@ -136,6 +139,23 @@ vertex.}
\label{fig:dual-depth} \label{fig:dual-depth}
\end{figure} \end{figure}
\begin{definition}[Depth-$d$ dual subgraph and its components]
\label{def:dual-component}
For $d \geq 0$, the \emph{depth-$d$ dual subgraph} is
\[
G'_d \;:=\; G'\bigl[\,\{d_f \in V(G') : \delta_G(d_f) = d\}\,\bigr],
\]
the inner-dual subgraph induced on the dual vertices of dual depth $d$.
For a connected component $C'$ of $G'_d$ we write
\[
F_{C'} := \{f : d_f \in V(C')\}, \qquad
V_{C'} := \bigcup_{f \in F_{C'}} V(f),
\]
for its set of faces and the vertices of $G$ bounding them, and
$R_{C'} := \bigcup_{f \in F_{C'}} f \subseteq |\Pi_G|$ for the closed
planar region these faces cover.
\end{definition}
\begin{definition}[Tire graph] \begin{definition}[Tire graph]
\label{def:tire-graph} \label{def:tire-graph}
A \emph{tire graph} consists of a plane graph $T$ together with an A \emph{tire graph} consists of a plane graph $T$ together with an
@@ -164,7 +184,9 @@ where $E_{\mathrm{ann}}$ --- the \emph{annular edges} --- has the
property that, in the plane embedding of $T$, the closed planar property that, in the plane embedding of $T$, the closed planar
region $R$ bounded externally by $B_{\mathrm{out}}$ and internally region $R$ bounded externally by $B_{\mathrm{out}}$ and internally
by $B_{\mathrm{in}}$ is partitioned into triangular faces of $T$ by $B_{\mathrm{in}}$ is partitioned into triangular faces of $T$
whose union is $R$. We call $R$ the \emph{tire tread} of $T$. whose union is $R$. We call $R$ the \emph{tire tread} of $T$ and write
$F_{\mathrm{ann}}$ for this set of triangular faces (the \emph{annular
faces}).
When $B_{\mathrm{out}}$ is a simple cycle and $O$ is $2$-connected, When $B_{\mathrm{out}}$ is a simple cycle and $O$ is $2$-connected,
the tread is a closed annulus. More generally, $R$ is a closed the tread is a closed annulus. More generally, $R$ is a closed
@@ -176,6 +198,11 @@ definition accommodates outerplanar inner graphs with bridges,
cut-vertices, or multiple connected components. When either cut-vertices, or multiple connected components. When either
boundary is degenerate, the tread is a closed disk with that vertex boundary is degenerate, the tread is a closed disk with that vertex
as apex. as apex.
We summarize the data of a tire graph as the triple
$T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$, from which $B_{\mathrm{in}}$,
the annular faces $F_{\mathrm{ann}}$, and the tread $R$ are determined;
we freely identify a tire graph with its underlying plane graph $T$.
\end{definition} \end{definition}
\begin{figure}[h] \begin{figure}[h]
@@ -193,12 +220,12 @@ triangular faces.}
\begin{remark} \begin{remark}
\label{rem:tire-counts} \label{rem:tire-counts}
Let $m = |V(B_{\mathrm{out}})|$ and $k = |V(B_{\mathrm{in}})|$. By Let $\mu = |V(B_{\mathrm{out}})|$ and $\nu = |V(B_{\mathrm{in}})|$. By
Euler's formula on the tire tread $R$, the tire graph has $m + k$ Euler's formula on the tire tread $R$, the tire graph has $\mu + \nu$
triangular faces inside $R$ and $|E_{\mathrm{ann}}| = m + k$ triangular faces inside $R$ and $|E_{\mathrm{ann}}| = \mu + \nu$
annular edges when neither boundary is degenerate; when exactly one annular edges when neither boundary is degenerate; when exactly one
boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$ boundary is degenerate (so $\min(\mu, \nu) = 1$), there are $\mu + \nu - 1$
triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$. triangular faces and $|E_{\mathrm{ann}}| = \mu + \nu - 1$.
\end{remark} \end{remark}
\begin{proposition}[Source-side simple-cycle property] \begin{proposition}[Source-side simple-cycle property]
@@ -268,16 +295,11 @@ contradicting the conclusion that the entire path lies on one side.
Let $G$ be a maximal planar graph and let $S \subseteq V(G)$ be a level Let $G$ be a maximal planar graph and let $S \subseteq V(G)$ be a level
source. Fix a plane embedding $\Pi_G$ of $G$ in which $S$ lies on the source. Fix a plane embedding $\Pi_G$ of $G$ in which $S$ lies on the
outer face (such an embedding exists for any planar graph and any outer face (such an embedding exists for any planar graph and any
single-vertex source). For $d \geq 0$, let single-vertex source). For $d \geq 0$, let $C'$ be a connected
\[ component of the depth-$d$ dual subgraph $G'_d$, with faces $F_{C'}$,
G'_d \;:=\; G'\bigl[\,\{d_f \in V(G') : \delta_G(d_f) = d\}\,\bigr] bounding vertices $V_{C'}$, and region $R_{C'}$ as in
\] Definition~\ref{def:dual-component}; let $C := G[V_{C'}]$ inherit its
be the inner-dual subgraph on dual vertices of dual depth $d$, and let embedding from $\Pi_G$.
$C'$ be a connected component of $G'_d$. Write
$F_{C'} := \{f : d_f \in V(C')\}$,
$V_{C'} := \bigcup_{f \in F_{C'}} V(f)$, and let $C := G[V_{C'}]$ inherit
its embedding from $\Pi_G$. Set $R_{C'} := \bigcup_{f \in F_{C'}} f
\subseteq |\Pi_G|$.
Then $C$, with the inherited embedding, is a tire graph in the sense of Definition~\ref{def:tire-graph}. Its outer boundary Then $C$, with the inherited embedding, is a tire graph in the sense of Definition~\ref{def:tire-graph}. 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 $\Pi_G$,
@@ -478,8 +500,11 @@ Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph,
and let $\Gamma$ be the graph on vertex set and let $\Gamma$ be the graph on vertex set
$\{d_f : f \in F_{\mathrm{ann}}\}$ with an edge $d_f d_{f'}$ for $\{d_f : f \in F_{\mathrm{ann}}\}$ with an edge $d_f d_{f'}$ for
each interior annular edge of $T$ (= each edge of $T$ whose two each interior annular edge of $T$ (= each edge of $T$ whose two
incident faces both lie in $F_{\mathrm{ann}}$). Then $\Gamma$ is incident faces both lie in $F_{\mathrm{ann}}$). Equivalently,
outerplanar. $\Gamma$ is the subgraph induced on $F_{\mathrm{ann}}$ of the
\emph{full tire dual} $D(T)$ --- the dual of $T$ taken over all of
its triangular faces, in which each boundary edge of $R$ contributes
a degree-$1$ vertex. Then $\Gamma$ is outerplanar.
Moreover, $\Gamma$ admits a planar embedding as a (possibly Moreover, $\Gamma$ admits a planar embedding as a (possibly
non-simple) Hamilton walk through every $d_f$, plus zero or more non-simple) Hamilton walk through every $d_f$, plus zero or more
@@ -569,12 +594,12 @@ $\partial R = B_{\mathrm{out}} \sqcup \overline{B_{\mathrm{in}}}$
appropriate orientation). Each boundary edge of $R$ is incident to appropriate orientation). Each boundary edge of $R$ is incident to
exactly one annular face: walking around $B_{\mathrm{out}}$ in exactly one annular face: walking around $B_{\mathrm{out}}$ in
cyclic order produces a sequence cyclic order produces a sequence
$f^{\mathrm{out}}_1, f^{\mathrm{out}}_2, \dots, f^{\mathrm{out}}_n$ $f^{\mathrm{out}}_1, f^{\mathrm{out}}_2, \dots, f^{\mathrm{out}}_\mu$
of (not necessarily distinct) annular faces, one per of (not necessarily distinct) annular faces, one per
$B_{\mathrm{out}}$-edge; similarly walking around $B_{\mathrm{out}}$-edge; similarly walking around
$B_{\mathrm{in}}$ produces a sequence $B_{\mathrm{in}}$ produces a sequence
$f^{\mathrm{in}}_1, \dots, f^{\mathrm{in}}_{m_\partial}$ where $f^{\mathrm{in}}_1, \dots, f^{\mathrm{in}}_{\nu_\partial}$ where
$m_\partial$ is the length of the inner-boundary walk. Pick any $\nu_\partial$ is the length of the inner-boundary walk. Pick any
spoke $e^\star = u w \in E_{\mathrm{ann}}$ with $u \in spoke $e^\star = u w \in E_{\mathrm{ann}}$ with $u \in
V(B_{\mathrm{out}})$ and $w \in V(B_{\mathrm{in}})$; cut $R$ along V(B_{\mathrm{out}})$ and $w \in V(B_{\mathrm{in}})$; cut $R$ along
$e^\star$. This converts the annulus into a closed disk $e^\star$. This converts the annulus into a closed disk
@@ -584,8 +609,8 @@ and once back along $e^\star$. Concatenating the two boundary
sequences (in the order dictated by this disk traversal) yields a sequences (in the order dictated by this disk traversal) yields a
single cyclic sequence single cyclic sequence
\[ \[
\mathcal{S} = (f^{\mathrm{out}}_1, \dots, f^{\mathrm{out}}_n, \mathcal{S} = (f^{\mathrm{out}}_1, \dots, f^{\mathrm{out}}_\mu,
f^{\mathrm{in}}_1, \dots, f^{\mathrm{in}}_{m_\partial}) f^{\mathrm{in}}_1, \dots, f^{\mathrm{in}}_{\nu_\partial})
\] \]
of annular faces with multiplicities. of annular faces with multiplicities.
@@ -742,7 +767,7 @@ $O$ $2$-connected and $E_{\mathrm{ann}}$ consisting only of spokes),
every annular face has exactly one boundary edge, every every annular face has exactly one boundary edge, every
$d_f$ has $\Gamma$-degree $2$, and the construction of the $d_f$ has $\Gamma$-degree $2$, and the construction of the
Theorem~\ref{thm:inner-dual-outerplanar} proof reduces to the Theorem~\ref{thm:inner-dual-outerplanar} proof reduces to the
classical Hamilton cycle $\Gamma \cong C_{n+m}$ with zero chords. classical Hamilton cycle $\Gamma \cong C_{\mu+\nu}$ with zero chords.
\end{remark} \end{remark}
\begin{remark} \begin{remark}
@@ -753,7 +778,7 @@ an interior annular edge in $\Gamma$ rather than two leaves in
$D(T)$ (see Definition~1.7 of \cite{bauerfeld-nested-tire-duals}). $D(T)$ (see Definition~1.7 of \cite{bauerfeld-nested-tire-duals}).
The two bridge-incident annular triangles have $\Gamma$-degree $3$; The two bridge-incident annular triangles have $\Gamma$-degree $3$;
the resulting $\Gamma$ has the structure of a Hamilton cycle of the resulting $\Gamma$ has the structure of a Hamilton cycle of
length $n + m_\partial$ plus a single chord (length $1$). This length $\mu + \nu_\partial$ plus a single chord (length $1$). This
corresponds to the theta graph $\Theta(1, b, c)$ identified corresponds to the theta graph $\Theta(1, b, c)$ identified
empirically in \cite{bauerfeld-nested-tire-duals}, which has no empirically in \cite{bauerfeld-nested-tire-duals}, which has no
$K_{2,3}$ subdivision (since one of the three paths has length $1$ $K_{2,3}$ subdivision (since one of the three paths has length $1$
@@ -823,13 +848,14 @@ stated equality.
\label{rem:count-general-outerplanar} \label{rem:count-general-outerplanar}
Theorem~\ref{thm:tait-tire} reduces the $4$-colouring count of a Theorem~\ref{thm:tait-tire} reduces the $4$-colouring count of a
tire to the $3$-edge-coloring count of its outerplanar inner dual tire to the $3$-edge-coloring count of its outerplanar inner dual
$\Gamma$. For the cycle case $\Gamma \cong C_n$ (the spoke-only $\Gamma$. For the cycle case $\Gamma \cong C_{\mu+\nu}$ (the spoke-only
case of Remark~\ref{rem:hamilton-cycle-spoke-only}), the cycle case of Remark~\ref{rem:hamilton-cycle-spoke-only}), the cycle
chromatic polynomial at $k = 3$ gives chromatic polynomial at $3$ colours gives
$2^n + 2 (-1)^n$. For an inner dual with one or more non-crossing $2^{\mu+\nu} + 2 (-1)^{\mu+\nu}$. For an inner dual with one or more
chords, the count depends on the chord structure, not just on the non-crossing chords, the count depends on the chord structure, not just
pair (number of vertices, number of chords): two outerplanar graphs on the pair (number of vertices, number of chords): two outerplanar
with the same $n$ and number of chords can have different proper graphs with the same number of vertices and number of chords can have
different proper
$3$-edge-coloring counts depending on how the chords are arranged $3$-edge-coloring counts depending on how the chords are arranged
(nested, sequential, sharing vertices, etc.). Every such count (nested, sequential, sharing vertices, etc.). Every such count
can nevertheless be computed in linear time by tree-decomposition can nevertheless be computed in linear time by tree-decomposition
@@ -851,23 +877,23 @@ tree structure $\mathcal{T}(G, S)$ defined as follows.
\item \emph{Root.} The depth-$0$ tire tread $T_0$ --- the unique \item \emph{Root.} The depth-$0$ tire tread $T_0$ --- the unique
tire produced by Lemma~\ref{lem:tire-component} at $d = 0$, tire produced by Lemma~\ref{lem:tire-component} at $d = 0$,
with degenerate outer boundary $B_{\mathrm{out}} = \{v_0\}$ with degenerate outer boundary $B_{\mathrm{out}} = \{v_0\}$
and inner outerplanar graph $O^{(0)} = G[L_1]$ --- is the and inner outerplanar graph $O^{(T_0)} = G[L_1]$ --- is the
root. root.
\item \emph{Parent.} For each tire tread $T_c$ at depth $d \ge 1$, \item \emph{Parent.} For each tire tread $T_c$ at depth $d \ge 1$,
its outer boundary $B_{\mathrm{out}}^{(c)}$ is a cycle in its outer boundary $B_{\mathrm{out}}^{(T_c)}$ is a cycle in
$L_d$. The \emph{parent} of $T_c$ is the unique tire tread $L_d$. The \emph{parent} of $T_c$ is the unique tire tread
$T_p$ at depth $d - 1$ whose inner outerplanar graph $T_p$ at depth $d - 1$ whose inner outerplanar graph
$O^{(p)}$ has $B_{\mathrm{out}}^{(c)}$ as the boundary cycle $O^{(T_p)}$ has $B_{\mathrm{out}}^{(T_c)}$ as the boundary cycle
of one of its bounded faces. Equivalently, $R_c$ lies of one of its bounded faces. Equivalently, $R_c$ lies
inside this bounded face of $O^{(p)}$ (which is itself the inside this bounded face of $O^{(T_p)}$ (which is itself the
region of the plane cut off by $B_{\mathrm{out}}^{(c)}$ on region of the plane cut off by $B_{\mathrm{out}}^{(T_c)}$ on
the side away from $S$). the side away from $S$).
\item \emph{Children.} The children of a tire tread $T_p$ are in \item \emph{Children.} The children of a tire tread $T_p$ are in
bijection with those bounded faces of $O^{(p)}$ whose bijection with those bounded faces of $O^{(T_p)}$ whose
interiors contain at least one vertex of $G$ at level interiors contain at least one vertex of $G$ at level
$\ge d + 2$ --- equivalently, with the connected components $\ge d + 2$ --- equivalently, with the connected components
of $G'_{d+1}$ whose tires have outer boundary cycle equal to of $G'_{d+1}$ whose tires have outer boundary cycle equal to
a bounded face of $O^{(p)}$. a bounded face of $O^{(T_p)}$.
\end{itemize} \end{itemize}
Every tire tread except $T_0$ has exactly one parent; a tire Every tire tread except $T_0$ has exactly one parent; a tire
@@ -884,34 +910,34 @@ gives the depth-$0$ tire $T_0$ described above.
\emph{Existence of parent.} Fix a tire tread $T_c$ at depth $d \emph{Existence of parent.} Fix a tire tread $T_c$ at depth $d
\ge 1$ arising from a connected component $C'_c$ of $G'_d$. Its \ge 1$ arising from a connected component $C'_c$ of $G'_d$. Its
outer boundary $B_{\mathrm{out}}^{(c)} = G[V_{C'_c} \cap L_d]$ is outer boundary $B_{\mathrm{out}}^{(T_c)} = G[V_{C'_c} \cap L_d]$ is
a simple cycle in $L_d$ (Lemma~\ref{lem:tire-component}; the a simple cycle in $L_d$ (Lemma~\ref{lem:tire-component}; the
source-side boundary of a tire is always a simple cycle, by source-side boundary of a tire is always a simple cycle, by
Proposition~\ref{prop:no-level-d-pinch}). The faces of $G$ Proposition~\ref{prop:no-level-d-pinch}). The faces of $G$
immediately outside $B_{\mathrm{out}}^{(c)}$ on the side facing $S$ immediately outside $B_{\mathrm{out}}^{(T_c)}$ on the side facing $S$
have depth $d - 1$ (one of their three vertices lies in $L_{d-1}$, have depth $d - 1$ (one of their three vertices lies in $L_{d-1}$,
two in $L_d$). Let $C'_p$ be the connected component of two in $L_d$). Let $C'_p$ be the connected component of
$G'_{d-1}$ containing the dual vertex of any such face. $G'_{d-1}$ containing the dual vertex of any such face.
\emph{Uniqueness of parent.} $B_{\mathrm{out}}^{(c)}$ is a single \emph{Uniqueness of parent.} $B_{\mathrm{out}}^{(T_c)}$ is a single
simple cycle in $G$, with a well-defined ``$S$-side'' (the side simple cycle in $G$, with a well-defined ``$S$-side'' (the side
of the cycle closer to $v_0$ in $\Pi_G$). The depth-$(d-1)$ of the cycle closer to $v_0$ in $\Pi_G$). The depth-$(d-1)$
faces lying on this side form a single contiguous arc around faces lying on this side form a single contiguous arc around
$B_{\mathrm{out}}^{(c)}$ in the dual --- they are all $G'$-adjacent $B_{\mathrm{out}}^{(T_c)}$ in the dual --- they are all $G'$-adjacent
in sequence (each pair of consecutive arc faces shares an edge in in sequence (each pair of consecutive arc faces shares an edge in
$B_{\mathrm{out}}^{(c)}$). Hence they all lie in the same $B_{\mathrm{out}}^{(T_c)}$). Hence they all lie in the same
connected component $C'_p$ of $G'_{d-1}$, which is therefore connected component $C'_p$ of $G'_{d-1}$, which is therefore
unique. unique.
\emph{$B_{\mathrm{out}}^{(c)}$ bounds a face of $O^{(p)}$.} The \emph{$B_{\mathrm{out}}^{(T_c)}$ bounds a face of $O^{(T_p)}$.} The
parent tire $T_p$ has $V(O^{(p)}) = V_{C'_p} \cap L_d \supseteq parent tire $T_p$ has $V(O^{(T_p)}) = V_{C'_p} \cap L_d \supseteq
V(B_{\mathrm{out}}^{(c)})$. The cycle $B_{\mathrm{out}}^{(c)}$ is V(B_{\mathrm{out}}^{(T_c)})$. The cycle $B_{\mathrm{out}}^{(T_c)}$ is
a subgraph of $O^{(p)}$ that bounds a face of $O^{(p)}$ in the a subgraph of $O^{(T_p)}$ that bounds a face of $O^{(T_p)}$ in the
inherited embedding: the cycle traces around a depth-$\ge d+1$ inherited embedding: the cycle traces around a depth-$\ge d+1$
region (containing $R_c$ and any descendants of $T_c$), which is region (containing $R_c$ and any descendants of $T_c$), which is
exactly a bounded face of $O^{(p)}$. exactly a bounded face of $O^{(T_p)}$.
\emph{Children description.} The bounded faces of $O^{(p)}$ are \emph{Children description.} The bounded faces of $O^{(T_p)}$ are
in bijection with the connected components of $G'_d$ whose in bijection with the connected components of $G'_d$ whose
faces lie inside those bounded regions (= one component per faces lie inside those bounded regions (= one component per
bounded face, by an argument analogous to the existence-and- bounded face, by an argument analogous to the existence-and-
@@ -926,12 +952,12 @@ decreases depth, terminating at $T_0$. No cycles can form
\begin{remark} \begin{remark}
\label{rem:tree-multiple-children} \label{rem:tree-multiple-children}
A parent tire $T_p$ has multiple children precisely when its A parent tire $T_p$ has multiple children precisely when its
inner outerplanar graph $O^{(p)}$ has multiple bounded faces with inner outerplanar graph $O^{(T_p)}$ has multiple bounded faces with
non-trivial interiors (= containing depth-$\ge d+2$ vertices of non-trivial interiors (= containing depth-$\ge d+2$ vertices of
$G$). This happens, for instance, when $O^{(p)}$ has chords or $G$). This happens, for instance, when $O^{(T_p)}$ has chords or
cut-vertices that subdivide its inner region, or when $O^{(p)}$ cut-vertices that subdivide its inner region, or when $O^{(T_p)}$
has multiple connected components in $G[L_{d+1}] \cap V_{C'_p}$. has multiple connected components in $G[L_{d+1}] \cap V_{C'_p}$.
By contrast, if $O^{(p)}$ is a simple cycle (the spoke-only case By contrast, if $O^{(T_p)}$ is a simple cycle (the spoke-only case
of Remark~\ref{rem:hamilton-cycle-spoke-only}) with a non-empty of Remark~\ref{rem:hamilton-cycle-spoke-only}) with a non-empty
interior, $T_p$ has exactly one child. interior, $T_p$ has exactly one child.
\end{remark} \end{remark}
@@ -1127,8 +1153,8 @@ proper coloring problem on $G$'s bounded faces factors through:
each tread is outerplanar by each tread is outerplanar by
Theorem~\ref{thm:inner-dual-outerplanar}), plus Theorem~\ref{thm:inner-dual-outerplanar}), plus
\item consistency constraints along parent-child interfaces (the \item consistency constraints along parent-child interfaces (the
cycle $B_{\mathrm{out}}^{(c)}$ shared between a child and the cycle $B_{\mathrm{out}}^{(T_c)}$ shared between a child and the
face of its parent's $O^{(p)}$). face of its parent's $O^{(T_p)}$).
\end{itemize} \end{itemize}
This is the structural setup underlying the chain-pigeonhole This is the structural setup underlying the chain-pigeonhole
program for tire treads. program for tire treads.