From 92f5efc3f1fa6d90c7df373b3221988519ba1a01 Mon Sep 17 00:00:00 2001 From: didericis Date: Fri, 29 May 2026 23:38:12 -0400 Subject: [PATCH] coloring_nested_tire_graphs: notation cleanup pass Define previously-implicit objects and unify conventions: - define level sets L_d (and 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 --- papers/coloring_nested_tire_graphs/paper.aux | 55 ++++---- papers/coloring_nested_tire_graphs/paper.log | 98 +++++++------ papers/coloring_nested_tire_graphs/paper.pdf | Bin 927226 -> 929392 bytes papers/coloring_nested_tire_graphs/paper.tex | 138 +++++++++++-------- 4 files changed, 165 insertions(+), 126 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index aaa0208..50d2f16 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -1,43 +1,43 @@ \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} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index c12cc92..342465f 100644 --- a/papers/coloring_nested_tire_graphs/paper.log +++ b/papers/coloring_nested_tire_graphs/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) 27 MAY 2026 22:18 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 29 MAY 2026 23:25 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -495,76 +495,88 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] - + File: fig_dual_depth.png Graphic file (type png) -Package pdftex.def Info: fig_dual_depth.png used on input line 128. +Package pdftex.def Info: fig_dual_depth.png used on input line 131. 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LaTeX Warning: `h' float specifier changed to `ht'. -[13] [14 <./fig_tire_tree_decomposition.png>] -Overfull \hbox (1.78508pt too wide) in paragraph at lines 1228--1230 +[14] [15 <./fig_tire_tree_decomposition.png>] +Overfull \hbox (1.78508pt too wide) in paragraph at lines 1254--1256 []\OT1/cmr/m/n/10 Length lower bound (Birkhoff). \OT1/cmr/m/it/10 Ev-ery non-tr ivial seam $\OML/cmm/m/it/10 C$ \OT1/cmr/m/it/10 of $\OML/cmm/m/it/10 G$ \OT1/c mr/m/it/10 has $\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 V\OT1/cmr/m/n/10 (\OML/cmm/m /it/10 C\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 j ^^U [] -[15] [16] (./paper.aux) ) +[16] (./paper.aux) ) Here is how much of TeX's memory you used: - 14061 strings out of 478268 - 279616 string characters out of 5846347 - 563910 words of memory out of 5000000 - 31884 multiletter control sequences out of 15000+600000 + 14062 strings out of 478268 + 279636 string characters out of 5846347 + 565983 words of memory out of 5000000 + 31885 multiletter control sequences out of 15000+600000 478218 words of font info for 62 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 84i,12n,89p,1168b,803s stack positions out of 10000i,1000n,20000p,200000b,200000s -< -/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb> -Output written on paper.pdf (16 pages, 927226 bytes). + + +Output written on paper.pdf (16 pages, 929392 bytes). 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a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index 2f64217..e1784c5 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -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 $L_{d}$, $L_{\leq d}$). \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.