diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 219ed4f..c091793 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -15,11 +15,10 @@ \newlabel{lem:tire-component}{{1.10}{5}} \citation{bauerfeld-pds} \citation{bauerfeld-pds} -\citation{Tait1880} \newlabel{rem:tire-component-degenerate}{{1.11}{7}} \newlabel{rem:tire-no-extra-hypotheses}{{1.12}{7}} -\newlabel{prop:tait-tire}{{1.13}{7}} -\newlabel{rem:tait}{{1.14}{7}} +\newlabel{def:complete-tire-dual}{{1.13}{7}} +\citation{Tait1880} \bibcite{Tait1880}{1} \bibcite{bauerfeld-pds}{2} \newlabel{tocindent-1}{0pt} @@ -27,5 +26,8 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} +\newlabel{prop:tait-tire-complete}{{1.14}{8}} +\newlabel{rem:tait-construction}{{1.15}{8}} +\newlabel{rem:tait-octahedron}{{1.16}{8}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{}\protected@file@percent } \gdef \@abspage@last{8} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index f1b4502..93cd840 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) 25 MAY 2026 18:48 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 18:59 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -215,10 +215,10 @@ Package pdftex.def Info: fig_partial_tire_dual.png used on input line 225. 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PDF statistics: 123 PDF objects out of 1000 (max. 8388607) 72 compressed objects within 1 object stream diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index 6f0b268..96fbc61 100644 Binary files a/papers/coloring_nested_tire_graphs/paper.pdf and b/papers/coloring_nested_tire_graphs/paper.pdf differ diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index 1a63966..03a9ef6 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -487,25 +487,71 @@ boundary cycle (the link of $v_0$); the corresponding tire graph has degenerate outer boundary $\{v_0\}$. \end{remark} -\begin{proposition}[Tait correspondence on the partial tire dual] -\label{prop:tait-tire} -The number of non-equivalent proper $4$-vertex-colorings of a tire -graph $T$ (modulo permutation of the four colors) equals the number -of non-equivalent proper $3$-edge-colorings of its partial tire dual -$D(T)$ (modulo permutation of the three colors). +\begin{definition}[Complete tire dual] +\label{def:complete-tire-dual} +The \emph{complete tire dual} $D^{\ast}(T)$ of a tire graph $T$ is +obtained from the partial tire dual $D(T)$ by quotienting its +leaves into non-annular-face vertices: +\begin{itemize} + \item Replace the $n$ outer leaves $\{\ell_e^{\mathrm{out}} : + e \in E(B_{\mathrm{out}})\}$ by a single \emph{outer-face + vertex} $v_{\mathrm{out}}$ of degree $n$. Each former leaf + edge $\{d_f, \ell_e^{\mathrm{out}}\}$ becomes an edge + $\{d_f, v_{\mathrm{out}}\}$ of $D^{\ast}(T)$. + \item For each bounded face $F$ of $O$ interior to $B_{\mathrm{in}}$, + replace the $|F|$ inner leaves whose edges lie on $\partial F$ + by a single \emph{inner-face vertex} $v_F$ of degree $|F|$, + in the same way. +\end{itemize} +Equivalently, $D^{\ast}(T)$ is the planar dual of $T$: one dual +vertex per face of $T$ (annular triangle, outer face, or bounded +interior face of $O$), one dual edge per edge of $T$. +\end{definition} + +\begin{proposition}[Tait correspondence on the complete tire dual] +\label{prop:tait-tire-complete} +Let $T$ be a tire graph. Then the number of non-equivalent proper +$4$-vertex-colorings of $T$ (modulo permutation of the four colors) +equals the number of non-equivalent \emph{Tait colorings} of +$D^{\ast}(T)$ (modulo permutation of the three nonzero elements of +$\mathbb{Z}_2 \times \mathbb{Z}_2$), where a Tait coloring is an +edge-labelling by the three nonzero elements of +$\mathbb{Z}_2 \times \mathbb{Z}_2$ such that at every vertex of +$D^{\ast}(T)$ the XOR of incident labels vanishes. \end{proposition} \begin{remark} -\label{rem:tait} -Proposition~\ref{prop:tait-tire} is the tire-graph analogue of Tait's -classical correspondence~\cite{Tait1880}: identifying the four colors -with the elements of $\mathbb{Z}_2 \times \mathbb{Z}_2$, the XOR of -the two endpoint colors of an edge of $T$ lies in the three nonzero -elements of $\mathbb{Z}_2 \times \mathbb{Z}_2$ and assigns a proper -$3$-edge-coloring to the corresponding edge of $D(T)$. The annular -triangles of $T$, encoded as the degree-$3$ vertices $d_f$ of $D(T)$, -contribute the requirement that each $d_f$'s three incident edges -carry three distinct colors. +\label{rem:tait-construction} +The correspondence is the classical Tait XOR map~\cite{Tait1880}: +identifying the four colors with $\mathbb{Z}_2 \times \mathbb{Z}_2$, +each edge of $T$ receives the XOR of its endpoint colors, which is +nonzero and so a Tait color. At any degree-$3$ vertex $d_f$ of +$D^{\ast}(T)$ (an annular triangle of $T$), the XOR-to-$0$ condition +forces the three incident edge colors to be distinct, recovering +proper $3$-edge-coloring at $d_f$. At the higher-degree dual +vertices $v_{\mathrm{out}}$ (degree $n$) and $v_F$ (degree $|F|$), +the XOR-to-$0$ condition is the non-trivial non-annular consistency +constraint absent from the partial tire dual $D(T)$. +\end{remark} + +\begin{remark} +\label{rem:tait-octahedron} +For the octahedron viewed as a tire graph with $n = m = 3$, +$O = C_3$ (no chords), and the spoke-only annular triangulation, +$D^{\ast}(T)$ is the cube $Q_3$: the six $d_f$ vertices form a +$6$-cycle, $v_{\mathrm{out}}$ is connected to the three O-move +$d_f$'s and $v_{\mathrm{in}}$ to the three I-move $d_f$'s, giving a +$3$-regular bipartite graph on $8$ vertices. The octahedron has +$96$ proper $4$-vertex-colorings, hence $96 / 24 = 4$ equivalence +classes modulo $S_4$; the cube has $24$ Tait colorings ($= 96 / 4$ +by Tait), hence $24 / 6 = 4$ equivalence classes modulo $S_3$. The +counts match, illustrating +Proposition~\ref{prop:tait-tire-complete}. In contrast, the partial +tire dual $D(T) \cong C_6 \circ K_1$ has $2^6 + 2 = 66$ proper +$3$-edge-colorings, i.e.\ $11$ classes modulo $S_3$, exceeding the +four vertex-coloring classes of $T$; the excess is precisely the +edge-colorings of $D(T)$ that violate the non-annular +XOR-to-$0$ constraints captured only in $D^{\ast}(T)$. \end{remark} \begin{thebibliography}{9}