From 6400fdfc5ef4e16fe5c7454bd41a51a81126854e Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 8 Jun 2026 15:09:58 -0400 Subject: [PATCH] Move tire coloring transfer to restrictions paper --- README.md | 4 +- .../paper.fdb_latexmk | 20 +- .../paper.fls | 36 ++-- .../paper.log | 4 +- .../paper.pdf | Bin 583446 -> 583433 bytes .../paper.tex | 2 +- .../check_inner_boundary_on_holton_mckay.py | 0 .../check_level_cycle_three_color.py | 0 .../draw_inner_boundary_counterexample.py | 0 .../experiments/draw_medial_tire_graph.py | 0 .../experiments/draw_seam_construction.py | 0 .../draw_tire_tree_decomposition.py | 0 ...aw_universal_level_cycle_counterexample.py | 0 .../generate_candidate_families.py | 0 .../experiments/kempe_classes.py | 0 .../experiments/level_cycle_support.py | 0 .../level_cycle_support_findings.md | 10 +- .../fig_dual_depth.png | Bin .../fig_inner_boundary_counterexample.png | Bin .../fig_medial_tire_example.png | Bin .../fig_seam_construction.png | Bin .../fig_tire_example.png | Bin .../fig_tire_tree_decomposition.png | Bin ...g_universal_level_cycle_counterexample.png | Bin .../paper.aux | 22 +-- .../paper.fdb_latexmk | 6 +- .../paper.fls | 2 +- .../paper.log | 84 ++++---- .../paper.pdf | Bin 1096120 -> 1082268 bytes .../paper.tex | 179 ----------------- .../paper.aux | 66 ++++--- .../paper.fdb_latexmk | 8 +- .../paper.fls | 3 + .../paper.log | 79 ++++---- .../paper.pdf | Bin 387544 -> 410946 bytes .../paper.tex | 183 +++++++++++++++++- 36 files changed, 372 insertions(+), 336 deletions(-) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/experiments/check_inner_boundary_on_holton_mckay.py (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/experiments/check_level_cycle_three_color.py (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/experiments/draw_inner_boundary_counterexample.py (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/experiments/draw_medial_tire_graph.py (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/experiments/draw_seam_construction.py (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/experiments/draw_tire_tree_decomposition.py (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/experiments/draw_universal_level_cycle_counterexample.py (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/experiments/generate_candidate_families.py (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/experiments/kempe_classes.py (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/experiments/level_cycle_support.py (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/experiments/level_cycle_support_findings.md (95%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/fig_dual_depth.png (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/fig_inner_boundary_counterexample.png (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/fig_medial_tire_example.png (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/fig_seam_construction.png (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/fig_tire_example.png (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/fig_tire_tree_decomposition.png (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/fig_universal_level_cycle_counterexample.png (100%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/paper.aux (90%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/paper.fdb_latexmk (98%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/paper.fls (99%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/paper.log (89%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/paper.pdf (86%) rename papers/{coloring_nested_tire_graphs => nested_tire_decompositions_of_plane_triangulations}/paper.tex (86%) diff --git a/README.md b/README.md index dd3750c..f2733c7 100644 --- a/README.md +++ b/README.md @@ -11,17 +11,17 @@ All papers are at `papers//paper.tex`. The current set: | `colored_edge_flip_classes` | Colored Edge Flip Classes | | `colored_pentagon_contractions` | Colored Pentagon Reductions | | `coloring_nested_tire_dual_graphs` | Coloring Nested Tire Dual Graphs | -| `coloring_nested_tire_graphs` | Coloring Nested Tire Graphs | | `even_level_graph_generators` | Even Level Graph Generators: a constructive conjecture stronger than the Four Color Theorem | | `face_monochromatic_pairs` | Face-Monochromatic Pairs and the Four Colour Theorem | | `iterated_reduction_in_reduced_dual` | An Iterated Reduction in the Reduced Dual | | `level_resolutions_of_maximal_planar_graphs` | Level Resolutions of Maximal Planar Graphs | | `level_switching` | Level Switching | +| `nested_tire_decompositions_of_plane_triangulations` | Nested Tire Decompositions of Plane Triangulations | | `plane_depth` | Plane Depth | | `plane_depth_sequencing` | Plane Depth Sequencing | | `plane_diamond_coloring` | Plane Diamond Coloring | -The papers form a connected programme around plane triangulations, BFS-level structure, and the Four Colour Theorem. `plane_depth` introduces the level / dual-depth framework that downstream papers build on; `coloring_nested_tire_graphs` develops the tire-tread tree-of-treads decomposition. +The papers form a connected programme around plane triangulations, BFS-level structure, and the Four Colour Theorem. `plane_depth` introduces the level / dual-depth framework that downstream papers build on; `nested_tire_decompositions_of_plane_triangulations` develops the tire-tread tree-of-treads decomposition. ## Creating a New Paper diff --git a/papers/coloring_nested_tire_dual_graphs/paper.fdb_latexmk b/papers/coloring_nested_tire_dual_graphs/paper.fdb_latexmk index 0923669..359a05f 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.fdb_latexmk +++ b/papers/coloring_nested_tire_dual_graphs/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1779735598 "paper.tex" "paper.pdf" "paper" 1779735599 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predicted. \end{remark} -\begin{theorem}[Tait correspondence: 4-colorings of a tire vs 3-edge-colorings of its inner dual] -\label{thm:tait-tire} -Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph -(viewed as an annular triangulation of its tire tread $R$) and let -$\Gamma$ be its inner dual -(Theorem~\ref{thm:inner-dual-outerplanar}). Then -\[ - \#\bigl\{\text{proper $4$-vertex-colorings of } T\bigr\} \big/ |S_4| - \;=\; - \#\bigl\{\text{proper $3$-edge-colorings of } \Gamma\bigr\} \big/ |S_3|. -\] -That is, the number of $4$-vertex-colorings of $T$ up to permutation -of the colour set $\{0, 1, 2, 3\}$ equals the number of -$3$-edge-colorings of $\Gamma$ up to permutation of the colour set -$\{1, 2, 3\}$. -\end{theorem} - -\begin{proof} -The argument is the classical Tait correspondence -\cite{tait-original} adapted to the annular triangulation $T$. -Encode the four colours of a proper $4$-vertex-coloring $c \colon -V(T) \to \mathbb{Z}_2 \times \mathbb{Z}_2$. For each interior -annular edge $e$ of $T$ (whose two incident faces both lie in -$F_{\mathrm{ann}}$, contributing a $\Gamma$-edge $e^*$), set -\[ - \chi^*(e^*) \;:=\; c(u) + c(v) \in \mathbb{Z}_2 \times \mathbb{Z}_2, - \qquad \text{where } u, v \text{ are the endpoints of } e. -\] -Since $c(u) \ne c(v)$, we have $\chi^*(e^*) \ne 00$, so $\chi^*$ -takes values in $\{01, 10, 11\}$, which we identify with the -$3$-edge-coloring palette $\{1, 2, 3\}$. - -\emph{Properness.} At each $\Gamma$-vertex $d_f$ corresponding to -an annular triangle $f = \{u, v, w\}$, the three incident -$\Gamma$-edges (one per cycle-edge of $f$) carry colours -$c(u) + c(v),\; c(v) + c(w),\; c(u) + c(w)$. These three elements -of $\mathbb{Z}_2 \times \mathbb{Z}_2$ sum to $0$ and are pairwise -distinct (their pairwise differences are -$c(u) - c(w),\; c(v) - c(u),\; c(w) - c(v)$, each nonzero), so -they form a permutation of $\{01, 10, 11\}$ --- a proper edge -colouring at $d_f$. - -\emph{Surjectivity onto cosets.} Given a proper $3$-edge-coloring -$\chi^*$ of $\Gamma$, the equation $c(u) + c(v) = \chi^*(e^*)$ -admits exactly $|\mathbb{Z}_2 \times \mathbb{Z}_2| = 4$ solutions -$c \colon V(T) \to \mathbb{Z}_2 \times \mathbb{Z}_2$ (a global -translation is the only freedom). Hence the map $c \mapsto -\chi^*$ is $4$-to-$1$. - -\emph{Count.} Therefore $\#\{\text{4-colorings of } T\} = 4 \cdot -\#\{\text{3-edge-colorings of } \Gamma\}$. Dividing by $|S_4| -= 24$ on the left and $|S_3| = 6$ on the right (since $S_4$ acts -faithfully on the $4$-colorings and $S_3$ on the $3$-edge-colorings, -and the $4$-to-$1$ map respects the $S_4/S_3 \cong S_3$ quotient -via the natural surjection $S_4 \twoheadrightarrow S_3$) gives the -stated equality. -\end{proof} - -\begin{remark} -\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 $\Gamma \cong C_{\mu+\nu}$ (the spoke-only -case of Remark~\ref{rem:hamilton-cycle-spoke-only}), the cycle -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 -methods, since outerplanar graphs have treewidth at most $2$ and -the edge-chromatic polynomial admits a deletion--contraction -recursion that respects the cycle-plus-chord structure. -\end{remark} - -\begin{definition}[Boundary-state chromatic transfer] -\label{def:boundary-state-transfer} -Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph. -Choose a cut along one annular edge if both boundaries are -non-degenerate; in the degenerate case make no cut. The tread becomes -a triangulated disk $\widetilde R$. Let -\[ - f_1, f_2, \ldots, f_m -\] -be any shelling order of the triangular faces of $\widetilde R$, i.e.\ -an order in which each initial union -$\widetilde R_i := f_1 \cup \cdots \cup f_i$ is a disk. Such an order -is obtained by taking an outerplanar embedding of the inner dual -$\Gamma$ from Theorem~\ref{thm:inner-dual-outerplanar} and repeatedly -removing an outer-face ear. - -For each $i$, let $A_i$ be the \emph{frontier}: the vertices of -$T$ incident to at least one processed face in $\widetilde R_i$ and to -at least one still-unprocessed constraint, where the unprocessed -constraints are the remaining annular faces together with any edge of -$O$ not yet tested by the transfer. A \emph{boundary state} on $A_i$ -is a partition $\pi$ of $A_i$ into colour classes, subject to the -condition that adjacent vertices of $T[A_i]$ lie in distinct blocks. -We write $r(\pi)$ for the number of blocks of $\pi$. -\end{definition} - -\begin{theorem}[Chromatic polynomial of a tire by frontier transfer] -\label{thm:tire-chromatic-polynomial-transfer} -For every tire graph $T$, the chromatic polynomial $P_T(q)$ is -computed by the following boundary-state dynamic program. - -Initialize the table at $i=0$ with the empty frontier state of weight -$1$. When the next triangular face -$f_i = \{x,y,z\}$ is attached, pass from states on $A_{i-1}$ to states -on $A_i$ as follows. -\begin{enumerate} -\item[(1)] Introduce any vertices of $f_i$ not already present in - $A_{i-1}$, assigning each such vertex either to an existing - colour block not containing one of its already-coloured - neighbours, or to a new block. -\item[(2)] Reject every assignment in which two adjacent vertices of - the triangle $f_i$ lie in the same block. Also reject every - assignment in which an edge of $O$ whose two endpoints have - now both appeared for the first time as a tested pair has - both endpoints in the same block. Thus chords and bridges of - the inner outerplanar graph are enforced exactly when their - second endpoint becomes visible to the transfer. -\item[(3)] Delete from the state every vertex no longer incident to an - unprocessed constraint. If deleting a vertex removes the last - representative of its colour block from the frontier, multiply - that transition by $1$; the colour has already been chosen. -\item[(4)] If a new vertex is assigned to a new colour block while the - current frontier state has $r$ colour blocks, multiply that - transition by $q-r$. If several new colour blocks are created - in the same triangle, the factors are - $(q-r)(q-r-1)\cdots$ in the order of creation. -\end{enumerate} -After $f_m$ is processed, the frontier is empty. The single remaining -weight is $P_T(q)$. -\end{theorem} - -\begin{proof} -The construction is the standard transfer for the chromatic polynomial, -specialized to the tire shelling. The frontier state records exactly -the equality pattern among colours that can still affect unprocessed -faces. Since colour names are irrelevant to the chromatic polynomial, -states are quotiented by the natural action of the symmetric group on -the colour set; a state with $r$ visible colour blocks can be extended -by a genuinely new colour in $q-r$ ways. - -Each transition accounts for all proper colourings of the enlarged -processed disk $\widetilde R_i$ that restrict to the resulting frontier -state, and accounts for none that violate an edge of the newly attached -triangle or untested edge of $O$. Vertices removed from the frontier -have no future incident unprocessed constraint, so their actual colour -names can no longer influence compatibility and may be forgotten. -Induction on $i$ therefore shows -that the table after step $i$ is precisely the orbit-count generating -function for proper colourings of $\widetilde R_i$ by frontier state. -At $i=m$ no vertices remain active, so the accumulated weight counts all -proper colourings of $T$. Because the weights are polynomials in $q$, -this count is the full chromatic polynomial. -\end{proof} - -\begin{remark}[Spoke-only transfer matrix] -\label{rem:spoke-only-chromatic-transfer} -In the spoke-only case with both boundaries simple cycles, the method -has a particularly small form. Cut the annulus along one spoke and -walk around the resulting strip. Each step adds one triangle sharing -an edge with the previous processed strip, so the frontier consists of -two or three consecutive boundary vertices. Up to colour permutation -there are only the possible equality patterns among those active -vertices, with adjacent vertices required to be distinct. The -chromatic polynomial is therefore the trace of a finite transfer matrix -whose entries are polynomials in $q$; the matrix depends only on the -local triangle type encountered while walking around the tread. Chords -or cut-vertices of $O$ enlarge the frontier only at the corresponding -outerplanar ears, and are handled by the same state rule of -Theorem~\ref{thm:tire-chromatic-polynomial-transfer}. -\end{remark} - \begin{theorem}[Tire treads form a rooted tree under face containment] \label{thm:tread-tree} Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ diff --git a/papers/three_color_restrictions_nested_tire_graphs/paper.aux b/papers/three_color_restrictions_nested_tire_graphs/paper.aux index 4a149ec..2f03a2b 100644 --- a/papers/three_color_restrictions_nested_tire_graphs/paper.aux +++ b/papers/three_color_restrictions_nested_tire_graphs/paper.aux @@ -13,40 +13,52 @@ \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Related work.}}{1}{}\protected@file@percent } \citation{bauerfeld-nested-tire-decompositions} +\citation{bauerfeld-nested-tire-decompositions} +\citation{tait-original} \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Background from nested tire decompositions}}{2}{}\protected@file@percent } -\newlabel{rem:level-cycle-motivation}{{2.1}{2}} -\newlabel{def:level-cycle-three-colour-restriction}{{2.2}{2}} -\newlabel{conj:false-universal-level-cycle-three-colour}{{2.3}{2}} -\newlabel{ex:universal-level-cycle-counterexample}{{2.4}{2}} -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The $8$-vertex counterexample to the universal-source form. 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Every triangulation in this range admits at least one vertex source witnessing the conjecture.}}{7}{}\protected@file@percent } +\newlabel{tab:level-cycle-three-colour-counts}{{1}{7}} +\citation{bauerfeld-nested-tire-decompositions} +\citation{bauerfeld-nested-tire-decompositions} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The $5$-connected slice at $n \leq 24$}}{8}{}\protected@file@percent } +\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces The $5$-connected triangulations at $14 \leq n \leq 24$ generated by \texttt {plantri -c5 -a}. 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In particular, a level source induces levels in a plane maximal planar graph, the depth-$d$ inner-dual components determine tire graphs, and the resulting tire treads form a rooted tire tree $\mathcal{T}(G,S)$. For a tread $T$, we write $B_{\mathrm{out}}^{(T)}$ and $B_{\mathrm{in}}^{(T)}$ for its outer and inner boundary data, $O^{(T)}$ for its inner outerplanar graph, and $G_T$ for the triangulated disk on the descendant side of $B_{\mathrm{out}}^{(T)}$. The base paper also records the boundary-state transfer viewpoint for a single tire and the factorisation of global colouring questions through local tread colourings together with compatibility along parent-child interfaces. +We use the terminology and structural results of~\cite{bauerfeld-nested-tire-decompositions}. In particular, a level source induces levels in a plane maximal planar graph, the depth-$d$ inner-dual components determine tire graphs, and the resulting tire treads form a rooted tire tree $\mathcal{T}(G,S)$. For a tread $T$, we write $B_{\mathrm{out}}^{(T)}$ and $B_{\mathrm{in}}^{(T)}$ for its outer and inner boundary data, $O^{(T)}$ for its inner outerplanar graph, and $G_T$ for the triangulated disk on the descendant side of $B_{\mathrm{out}}^{(T)}$. The base paper also proves that each tire tread has an outerplanar inner dual and that global colouring questions factor through local tread colourings together with compatibility along parent-child interfaces. + +\section{Single-tire colouring transfer} + +\begin{theorem}[Tait correspondence: 4-colorings of a tire vs 3-edge-colorings of its inner dual] +\label{thm:tait-tire} +Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph +(viewed as an annular triangulation of its tire tread $R$) and let +$\Gamma$ be its outerplanar inner dual, as supplied by +\cite{bauerfeld-nested-tire-decompositions}. Then +\[ + \#\bigl\{\text{proper $4$-vertex-colorings of } T\bigr\} \big/ |S_4| + \;=\; + \#\bigl\{\text{proper $3$-edge-colorings of } \Gamma\bigr\} \big/ |S_3|. +\] +That is, the number of $4$-vertex-colorings of $T$ up to permutation +of the colour set $\{0, 1, 2, 3\}$ equals the number of +$3$-edge-colorings of $\Gamma$ up to permutation of the colour set +$\{1, 2, 3\}$. +\end{theorem} + +\begin{proof} +The argument is the classical Tait correspondence +\cite{tait-original} adapted to the annular triangulation $T$. +Encode the four colours of a proper $4$-vertex-coloring $c \colon +V(T) \to \mathbb{Z}_2 \times \mathbb{Z}_2$. For each interior +annular edge $e$ of $T$ (whose two incident faces both lie in +$F_{\mathrm{ann}}$, contributing a $\Gamma$-edge $e^*$), set +\[ + \chi^*(e^*) \;:=\; c(u) + c(v) \in \mathbb{Z}_2 \times \mathbb{Z}_2, + \qquad \text{where } u, v \text{ are the endpoints of } e. +\] +Since $c(u) \ne c(v)$, we have $\chi^*(e^*) \ne 00$, so $\chi^*$ +takes values in $\{01, 10, 11\}$, which we identify with the +$3$-edge-coloring palette $\{1, 2, 3\}$. + +\emph{Properness.} At each $\Gamma$-vertex $d_f$ corresponding to +an annular triangle $f = \{u, v, w\}$, the three incident +$\Gamma$-edges (one per cycle-edge of $f$) carry colours +$c(u) + c(v),\; c(v) + c(w),\; c(u) + c(w)$. These three elements +of $\mathbb{Z}_2 \times \mathbb{Z}_2$ sum to $0$ and are pairwise +distinct (their pairwise differences are +$c(u) - c(w),\; c(v) - c(u),\; c(w) - c(v)$, each nonzero), so +they form a permutation of $\{01, 10, 11\}$ --- a proper edge +colouring at $d_f$. + +\emph{Surjectivity onto cosets.} Given a proper $3$-edge-coloring +$\chi^*$ of $\Gamma$, the equation $c(u) + c(v) = \chi^*(e^*)$ +admits exactly $|\mathbb{Z}_2 \times \mathbb{Z}_2| = 4$ solutions +$c \colon V(T) \to \mathbb{Z}_2 \times \mathbb{Z}_2$ (a global +translation is the only freedom). Hence the map $c \mapsto +\chi^*$ is $4$-to-$1$. + +\emph{Count.} Therefore $\#\{\text{4-colorings of } T\} = 4 \cdot +\#\{\text{3-edge-colorings of } \Gamma\}$. Dividing by $|S_4| += 24$ on the left and $|S_3| = 6$ on the right (since $S_4$ acts +faithfully on the $4$-colorings and $S_3$ on the $3$-edge-colorings, +and the $4$-to-$1$ map respects the $S_4/S_3 \cong S_3$ quotient +via the natural surjection $S_4 \twoheadrightarrow S_3$) gives the +stated equality. +\end{proof} + +\begin{remark} +\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 $\Gamma \cong C_{\mu+\nu}$ (the spoke-only +case described in \cite{bauerfeld-nested-tire-decompositions}), the cycle +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 +methods, since outerplanar graphs have treewidth at most $2$ and +the edge-chromatic polynomial admits a deletion--contraction +recursion that respects the cycle-plus-chord structure. +\end{remark} + +\begin{definition}[Boundary-state chromatic transfer] +\label{def:boundary-state-transfer} +Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph. +Choose a cut along one annular edge if both boundaries are +non-degenerate; in the degenerate case make no cut. The tread becomes +a triangulated disk $\widetilde R$. Let +\[ + f_1, f_2, \ldots, f_m +\] +be any shelling order of the triangular faces of $\widetilde R$, i.e.\ +an order in which each initial union +$\widetilde R_i := f_1 \cup \cdots \cup f_i$ is a disk. Such an order +is obtained by taking an outerplanar embedding of the inner dual +$\Gamma$ from \cite{bauerfeld-nested-tire-decompositions} and repeatedly +removing an outer-face ear. + +For each $i$, let $A_i$ be the \emph{frontier}: the vertices of +$T$ incident to at least one processed face in $\widetilde R_i$ and to +at least one still-unprocessed constraint, where the unprocessed +constraints are the remaining annular faces together with any edge of +$O$ not yet tested by the transfer. A \emph{boundary state} on $A_i$ +is a partition $\pi$ of $A_i$ into colour classes, subject to the +condition that adjacent vertices of $T[A_i]$ lie in distinct blocks. +We write $r(\pi)$ for the number of blocks of $\pi$. +\end{definition} + +\begin{theorem}[Chromatic polynomial of a tire by frontier transfer] +\label{thm:tire-chromatic-polynomial-transfer} +For every tire graph $T$, the chromatic polynomial $P_T(q)$ is +computed by the following boundary-state dynamic program. + +Initialize the table at $i=0$ with the empty frontier state of weight +$1$. When the next triangular face +$f_i = \{x,y,z\}$ is attached, pass from states on $A_{i-1}$ to states +on $A_i$ as follows. +\begin{enumerate} +\item[(1)] Introduce any vertices of $f_i$ not already present in + $A_{i-1}$, assigning each such vertex either to an existing + colour block not containing one of its already-coloured + neighbours, or to a new block. +\item[(2)] Reject every assignment in which two adjacent vertices of + the triangle $f_i$ lie in the same block. Also reject every + assignment in which an edge of $O$ whose two endpoints have + now both appeared for the first time as a tested pair has + both endpoints in the same block. Thus chords and bridges of + the inner outerplanar graph are enforced exactly when their + second endpoint becomes visible to the transfer. +\item[(3)] Delete from the state every vertex no longer incident to an + unprocessed constraint. If deleting a vertex removes the last + representative of its colour block from the frontier, multiply + that transition by $1$; the colour has already been chosen. +\item[(4)] If a new vertex is assigned to a new colour block while the + current frontier state has $r$ colour blocks, multiply that + transition by $q-r$. If several new colour blocks are created + in the same triangle, the factors are + $(q-r)(q-r-1)\cdots$ in the order of creation. +\end{enumerate} +After $f_m$ is processed, the frontier is empty. The single remaining +weight is $P_T(q)$. +\end{theorem} + +\begin{proof} +The construction is the standard transfer for the chromatic polynomial, +specialized to the tire shelling. The frontier state records exactly +the equality pattern among colours that can still affect unprocessed +faces. Since colour names are irrelevant to the chromatic polynomial, +states are quotiented by the natural action of the symmetric group on +the colour set; a state with $r$ visible colour blocks can be extended +by a genuinely new colour in $q-r$ ways. + +Each transition accounts for all proper colourings of the enlarged +processed disk $\widetilde R_i$ that restrict to the resulting frontier +state, and accounts for none that violate an edge of the newly attached +triangle or untested edge of $O$. Vertices removed from the frontier +have no future incident unprocessed constraint, so their actual colour +names can no longer influence compatibility and may be forgotten. +Induction on $i$ therefore shows +that the table after step $i$ is precisely the orbit-count generating +function for proper colourings of $\widetilde R_i$ by frontier state. +At $i=m$ no vertices remain active, so the accumulated weight counts all +proper colourings of $T$. Because the weights are polynomials in $q$, +this count is the full chromatic polynomial. +\end{proof} + +\begin{remark}[Spoke-only transfer matrix] +\label{rem:spoke-only-chromatic-transfer} +In the spoke-only case with both boundaries simple cycles, the method +has a particularly small form. Cut the annulus along one spoke and +walk around the resulting strip. Each step adds one triangle sharing +an edge with the previous processed strip, so the frontier consists of +two or three consecutive boundary vertices. Up to colour permutation +there are only the possible equality patterns among those active +vertices, with adjacent vertices required to be distinct. The +chromatic polynomial is therefore the trace of a finite transfer matrix +whose entries are polynomials in $q$; the matrix depends only on the +local triangle type encountered while walking around the tread. Chords +or cut-vertices of $O$ enlarge the frontier only at the corresponding +outerplanar ears, and are handled by the same state rule of +Theorem~\ref{thm:tire-chromatic-polynomial-transfer}. +\end{remark} \begin{remark}[Motivation for level-cycle restrictions] \label{rem:level-cycle-motivation}