Move tire coloring transfer to restrictions paper
This commit is contained in:
@@ -11,17 +11,17 @@ All papers are at `papers/<name>/paper.tex`. The current set:
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| `colored_edge_flip_classes` | Colored Edge Flip Classes |
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| `colored_pentagon_contractions` | Colored Pentagon Reductions |
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| `coloring_nested_tire_dual_graphs` | Coloring Nested Tire Dual Graphs |
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| `coloring_nested_tire_graphs` | Coloring Nested Tire Graphs |
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| `even_level_graph_generators` | Even Level Graph Generators: a constructive conjecture stronger than the Four Color Theorem |
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| `face_monochromatic_pairs` | Face-Monochromatic Pairs and the Four Colour Theorem |
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| `iterated_reduction_in_reduced_dual` | An Iterated Reduction in the Reduced Dual |
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| `level_resolutions_of_maximal_planar_graphs` | Level Resolutions of Maximal Planar Graphs |
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| `level_switching` | Level Switching |
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| `nested_tire_decompositions_of_plane_triangulations` | Nested Tire Decompositions of Plane Triangulations |
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| `plane_depth` | Plane Depth |
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| `plane_depth_sequencing` | Plane Depth Sequencing |
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| `plane_diamond_coloring` | Plane Diamond Coloring |
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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.
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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.
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## Creating a New Paper
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@@ -1,5 +1,5 @@
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# Fdb version 3
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"paper.aux"
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"paper.log"
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@@ -1,4 +1,4 @@
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PWD /Users/didericis/Code/math-research/papers/coloring_nested_tire_graphs
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PWD /Users/didericis/Code/math-research/papers/coloring_nested_tire_dual_graphs
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INPUT ./fig_dual_depth.png
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INPUT ./fig_dual_depth.png
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INPUT fig_dual_depth.png
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INPUT ./fig_dual_depth.png
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OUTPUT paper.pdf
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INPUT ./fig_dual_depth.png
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INPUT /usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map
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INPUT ./fig_tire_example.png
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INPUT ./fig_tire_example.png
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INPUT fig_tire_example.png
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INPUT ./fig_tire_example.png
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INPUT ./fig_tire_example.png
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INPUT ./fig_partial_tire_dual.png
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INPUT ./fig_partial_tire_dual.png
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INPUT fig_partial_tire_dual.png
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INPUT ./fig_partial_tire_dual.png
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INPUT ./fig_partial_tire_dual.png
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INPUT ./fig_partial_tire_dual_bridge.png
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INPUT ./fig_partial_tire_dual_bridge.png
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INPUT fig_partial_tire_dual_bridge.png
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INPUT ./fig_partial_tire_dual_bridge.png
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INPUT ./fig_partial_tire_dual_bridge.png
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INPUT ./notes/fig_facial_dual_choices.png
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INPUT ./notes/fig_facial_dual_choices.png
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INPUT notes/fig_facial_dual_choices.png
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INPUT ./notes/fig_facial_dual_choices.png
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Output written on paper.pdf (7 pages, 583446 bytes).
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Output written on paper.pdf (7 pages, 583433 bytes).
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@@ -469,7 +469,7 @@ manuscript (math-research repository), 2026.
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\bibitem{bauerfeld-nested-tires}
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E.~Bauerfeld,
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\emph{Coloring Nested Tire Graphs},
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\emph{Nested Tire Decompositions of Plane Triangulations},
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manuscript (math-research repository), 2026.
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\end{thebibliography}
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+5
-5
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Executable:
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```bash
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python3 papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py
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python3 papers/nested_tire_decompositions_of_plane_triangulations/experiments/level_cycle_support.py
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```
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## Model
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@@ -35,7 +35,7 @@ of `C`.
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Command:
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```bash
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python3 -u papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py \
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python3 -u papers/nested_tire_decompositions_of_plane_triangulations/experiments/level_cycle_support.py \
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--n-min 3 --n-max 6 --outer-max 5 --inner-max 7 --max-chords 2 --max-paths 30
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```
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@@ -133,7 +133,7 @@ window for `n = 6`. The smallest admissible inner ring is `k = 7`
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Command:
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```bash
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python3 -u papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py \
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python3 -u papers/nested_tire_decompositions_of_plane_triangulations/experiments/level_cycle_support.py \
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--n-min 6 --n-max 6 --outer-min 3 --outer-max 6 --inner-min 7 --inner-max 8 \
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--progress --examples 6
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```
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@@ -175,7 +175,7 @@ strictly *less* restrictive supports — never a new floor.
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Command:
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```bash
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python3 -u papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py \
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python3 -u papers/nested_tire_decompositions_of_plane_triangulations/experiments/level_cycle_support.py \
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--n-min 6 --n-max 6 --outer-min 3 --outer-max 7 --inner-min 7 --inner-max 9 \
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--progress --examples 6
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```
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@@ -246,7 +246,7 @@ To probe the conjecture more directly we ran a "skip-`m=3`" sweep
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explicitly excluding `m = 3` from the search:
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```bash
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python3 -u papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py \
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python3 -u papers/nested_tire_decompositions_of_plane_triangulations/experiments/level_cycle_support.py \
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--n-min 6 --n-max 6 --outer-min 4 --outer-max 10 --inner-min 7 --inner-max 9 \
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--progress
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```
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\newlabel{fig:inner-dual-annulus-case}{{5}{11}}
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\newlabel{rem:hamilton-cycle-spoke-only}{{1.16}{11}}
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\newlabel{rem:bridge-case-theta}{{1.17}{11}}
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\citation{tait-original}
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\newlabel{thm:tait-tire}{{1.18}{12}}
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\newlabel{rem:count-general-outerplanar}{{1.19}{12}}
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\newlabel{def:boundary-state-transfer}{{1.20}{13}}
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\newlabel{thm:tire-chromatic-polynomial-transfer}{{1.21}{13}}
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\newlabel{rem:spoke-only-chromatic-transfer}{{1.22}{14}}
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\newlabel{thm:tread-tree}{{1.23}{14}}
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\newlabel{rem:tree-multiple-children}{{1.24}{15}}
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\newlabel{thm:tire-tree-decomposition}{{1.25}{15}}
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\newlabel{thm:tread-tree}{{1.18}{12}}
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\newlabel{rem:tree-multiple-children}{{1.19}{13}}
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\newlabel{thm:tire-tree-decomposition}{{1.20}{13}}
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\@writefile{lof}{\contentsline {figure}{\numberline {6}{\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 }
|
||||
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|
||||
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|
||||
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|
||||
\bibcite{bauerfeld-depth}{2}
|
||||
\bibcite{bauerfeld-nested-tire-duals}{3}
|
||||
@@ -65,8 +62,5 @@
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\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.25\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 $(a)$.}}{18}{}\protected@file@percent }
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-179
@@ -894,185 +894,6 @@ and so contributes no degree-$2$ branch vertex), hence is
|
||||
outerplanar as 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 $\Pi_G$
|
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@@ -114,7 +114,188 @@ organised around a global nested-cycle decomposition of this kind.
|
||||
|
||||
\section{Background from nested tire decompositions}
|
||||
|
||||
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 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}
|
||||
|
||||
Reference in New Issue
Block a user