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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
\citation{bauerfeld-nested-tire-duals}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
\newlabel{def:dual}{{1.3}{1}}
\newlabel{def:dual}{{1.3}{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 }
\newlabel{fig:dual-depth}{{1}{2}}
\newlabel{def:tire-graph}{{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{fig:tire-example}{{2}{3}}
\newlabel{rem:tire-counts}{{1.6}{3}}
\newlabel{prop:no-level-d-pinch}{{1.7}{3}}
\newlabel{lem:tire-component}{{1.8}{4}}
\newlabel{def:dual-component}{{1.5}{2}}
\newlabel{def:tire-graph}{{1.6}{3}}
\newlabel{rem:tire-counts}{{1.7}{3}}
\newlabel{prop:no-level-d-pinch}{{1.8}{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}}
\citation{bauerfeld-depth}
\newlabel{lem:tire-component}{{1.9}{5}}
\citation{bauerfeld-depth}
\newlabel{thm:tread-partition}{{1.9}{6}}
\newlabel{rem:tire-component-degenerate}{{1.10}{6}}
\newlabel{rem:tire-no-extra-hypotheses}{{1.11}{6}}
\newlabel{thm:inner-dual-outerplanar}{{1.12}{7}}
\newlabel{thm:tread-partition}{{1.10}{6}}
\newlabel{rem:tire-component-degenerate}{{1.11}{7}}
\newlabel{rem:tire-no-extra-hypotheses}{{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 }
\newlabel{fig:inner-dual-disk-case}{{3}{8}}
\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}
\newlabel{thm:tait-tire}{{1.15}{10}}
\newlabel{rem:count-general-outerplanar}{{1.16}{10}}
\newlabel{thm:tread-tree}{{1.17}{10}}
\newlabel{rem:tree-multiple-children}{{1.18}{11}}
\newlabel{thm:tire-tree-decomposition}{{1.19}{12}}
\newlabel{rem:tree-coloring-factorisation}{{1.20}{13}}
\@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{fig:tire-tree-decomposition}{{5}{14}}
\newlabel{def:seam}{{1.21}{14}}
\newlabel{def:partial-tire-tree}{{1.22}{15}}
\newlabel{lem:seam-edge-shared}{{1.23}{15}}
\newlabel{conj:seam-counterexample}{{1.24}{15}}
\newlabel{rem:hamilton-cycle-spoke-only}{{1.14}{9}}
\newlabel{rem:bridge-case-theta}{{1.15}{9}}
\newlabel{thm:tait-tire}{{1.16}{9}}
\@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{fig:inner-dual-annulus-case}{{4}{10}}
\newlabel{rem:count-general-outerplanar}{{1.17}{11}}
\newlabel{thm:tread-tree}{{1.18}{11}}
\newlabel{rem:tree-multiple-children}{{1.19}{12}}
\newlabel{thm:tire-tree-decomposition}{{1.20}{12}}
\newlabel{rem:tree-coloring-factorisation}{{1.21}{14}}
\newlabel{def:seam}{{1.22}{14}}
\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{bauerfeld-depth}{2}
\bibcite{bauerfeld-nested-tire-duals}{3}
@@ -46,5 +46,6 @@
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\newlabel{conj:seam-counterexample}{{1.25}{16}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{16}{}\protected@file@percent }
\gdef \@abspage@last{16}
papers/coloring_nested_tire_graphs/paper.log
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[]\OT1/cmr/m/n/10 Length lower bound (Birkhoff). \OT1/cmr/m/it/10 Ev-ery non-tr
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papers/coloring_nested_tire_graphs/paper.tex
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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]
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.
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}
\begin{definition}[Dual]
@@ -136,6 +139,23 @@ vertex.}
\label{fig:dual-depth}
\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]
\label{def:tire-graph}
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
region $R$ bounded externally by $B_{\mathrm{out}}$ and internally
by $B_{\mathrm{in}}$ is partitioned into triangular faces of $T$
whose union is $R$. 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,
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
boundary is degenerate, the tread is a closed disk with that vertex
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}
\begin{figure}[h]
@@ -193,12 +220,12 @@ triangular faces.}
\begin{remark}
\label{rem:tire-counts}
Let $m = |V(B_{\mathrm{out}})|$ and $k = |V(B_{\mathrm{in}})|$. By
Euler's formula on the tire tread $R$, the tire graph has $m + k$
triangular faces inside $R$ and $|E_{\mathrm{ann}}| = m + k$
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 $\mu + \nu$
triangular faces inside $R$ and $|E_{\mathrm{ann}}| = \mu + \nu$
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 $|E_{\mathrm{ann}}| = m + k - 1$.
boundary is degenerate (so $\min(\mu, \nu) = 1$), there are $\mu + \nu - 1$
triangular faces and $|E_{\mathrm{ann}}| = \mu + \nu - 1$.
\end{remark}
\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
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
single-vertex source). For $d \geq 0$, let
\[
G'_d \;:=\; G'\bigl[\,\{d_f \in V(G') : \delta_G(d_f) = d\}\,\bigr]
\]
be the inner-dual subgraph on dual vertices of dual depth $d$, and let
$C'$ 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 $R_{C'} := \bigcup_{f \in F_{C'}} f
\subseteq |\Pi_G|$.
single-vertex source). For $d \geq 0$, let $C'$ be a connected
component of the depth-$d$ dual subgraph $G'_d$, with faces $F_{C'}$,
bounding vertices $V_{C'}$, and region $R_{C'}$ as in
Definition~\ref{def:dual-component}; let $C := G[V_{C'}]$ inherit its
embedding from $\Pi_G$.
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 $\Pi_G$,
@@ -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
$\{d_f : f \in F_{\mathrm{ann}}\}$ with an edge $d_f d_{f'}$ for
each interior annular edge of $T$ (= each edge of $T$ whose two
incident faces both lie in $F_{\mathrm{ann}}$). Then $\Gamma$ is
outerplanar.
incident faces both lie in $F_{\mathrm{ann}}$). Equivalently,
$\Gamma$ 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
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
exactly one annular face: walking around $B_{\mathrm{out}}$ in
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
$B_{\mathrm{out}}$-edge; similarly walking around
$B_{\mathrm{in}}$ produces a sequence
$f^{\mathrm{in}}_1, \dots, f^{\mathrm{in}}_{m_\partial}$ where
$m_\partial$ is the length of the inner-boundary walk. Pick any
$f^{\mathrm{in}}_1, \dots, f^{\mathrm{in}}_{\nu_\partial}$ where
$\nu_\partial$ is the length of the inner-boundary walk. Pick any
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
$e^\star$. 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
single cyclic sequence
\[
\mathcal{S} = (f^{\mathrm{out}}_1, \dots, f^{\mathrm{out}}_n,
f^{\mathrm{in}}_1, \dots, f^{\mathrm{in}}_{m_\partial})
\mathcal{S} = (f^{\mathrm{out}}_1, \dots, f^{\mathrm{out}}_\mu,
f^{\mathrm{in}}_1, \dots, f^{\mathrm{in}}_{\nu_\partial})
\]
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
$d_f$ has $\Gamma$-degree $2$, and the construction of 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}
\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}).
The two bridge-incident annular triangles have $\Gamma$-degree $3$;
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
empirically in \cite{bauerfeld-nested-tire-duals}, which has no
$K_{2,3}$ subdivision (since one of the three paths has length $1$
@@ -823,13 +848,14 @@ stated equality.
\label{rem:count-general-outerplanar}
Theorem~\ref{thm:tait-tire} reduces the $4$-colouring count of a
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
chromatic polynomial at $k = 3$ gives
$2^n + 2 (-1)^n$. For an inner dual with one or more non-crossing
chords, the count depends on the chord structure, not just on the
pair (number of vertices, number of chords): two outerplanar graphs
with the same $n$ and number of chords can have different proper
chromatic polynomial at $3$ colours gives
$2^{\mu+\nu} + 2 (-1)^{\mu+\nu}$. For an inner dual with one or more
non-crossing chords, the count depends on the chord structure, not just
on the pair (number of vertices, number of chords): two outerplanar
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
(nested, sequential, sharing vertices, etc.). Every such count
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
tire produced by Lemma~\ref{lem:tire-component} at $d = 0$,
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.
\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
$T_p$ 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
inside this bounded face of $O^{(p)}$ (which is itself the
region of the plane cut off by $B_{\mathrm{out}}^{(c)}$ on
inside this bounded face of $O^{(T_p)}$ (which is itself the
region of the plane cut off by $B_{\mathrm{out}}^{(T_c)}$ on
the side away from $S$).
\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
$\ge d + 2$ --- equivalently, with the connected components
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}
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 $d
\ge 1$ 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
source-side boundary of a tire is always a simple cycle, by
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 $L_{d-1}$,
two in $L_d$). Let $C'_p$ be the connected component of
$G'_{d-1}$ 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
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
$B_{\mathrm{out}}^{(c)}$ 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
$B_{\mathrm{out}}^{(c)}$). 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
unique.
\emph{$B_{\mathrm{out}}^{(c)}$ bounds a face of $O^{(p)}$.} The
parent tire $T_p$ has $V(O^{(p)}) = V_{C'_p} \cap L_d \supseteq
V(B_{\mathrm{out}}^{(c)})$. The cycle $B_{\mathrm{out}}^{(c)}$ is
a subgraph of $O^{(p)}$ that bounds a face of $O^{(p)}$ in the
\emph{$B_{\mathrm{out}}^{(T_c)}$ bounds a face of $O^{(T_p)}$.} The
parent tire $T_p$ has $V(O^{(T_p)}) = V_{C'_p} \cap L_d \supseteq
V(B_{\mathrm{out}}^{(T_c)})$. The cycle $B_{\mathrm{out}}^{(T_c)}$ is
a subgraph of $O^{(T_p)}$ that bounds a face of $O^{(T_p)}$ in the
inherited embedding: the cycle traces around a depth-$\ge d+1$
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
faces lie inside those bounded regions (= one component per
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}
\label{rem:tree-multiple-children}
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
$G$). This happens, for instance, when $O^{(p)}$ has chords or
cut-vertices that subdivide its inner region, or when $O^{(p)}$
$G$). This happens, for instance, when $O^{(T_p)}$ has chords or
cut-vertices that subdivide its inner region, or when $O^{(T_p)}$
has multiple connected components in $G[L_{d+1}] \cap V_{C'_p}$.
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
interior, $T_p$ has exactly one child.
\end{remark}
@@ -1127,8 +1153,8 @@ proper coloring problem on $G$'s bounded faces factors through:
each tread is outerplanar by
Theorem~\ref{thm:inner-dual-outerplanar}), plus
\item consistency constraints along parent-child interfaces (the
cycle $B_{\mathrm{out}}^{(c)}$ shared between a child and the
face of its parent's $O^{(p)}$).
cycle $B_{\mathrm{out}}^{(T_c)}$ shared between a child and the
face of its parent's $O^{(T_p)}$).
\end{itemize}
This is the structural setup underlying the chain-pigeonhole
program for tire treads.