From 35218f21919cbad90eb36e30fc8e271bc51a3220 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 18:48:10 -0400 Subject: [PATCH] coloring_nested_tire_graphs: state Tait correspondence on partial tire dual; cite Tait 1880 Adds Proposition 1.13: the number of non-equivalent proper 4-vertex- colorings of a tire graph T (mod S_4) equals the number of non- equivalent proper 3-edge-colorings of its partial tire dual D(T) (mod S_3). The map is the classical Tait XOR construction: identifying the four colors with Z_2 x Z_2, each edge of T receives an edge color equal to the XOR of its endpoint colors, which lies in the three nonzero elements of Z_2 x Z_2 -- giving the corresponding edge of D(T) a 3-edge-color. Annular triangles of T, encoded as degree-3 vertices d_f of D(T), supply the three-distinct-colors constraint. Adds Remark 1.14 explaining the analogy with Tait's classical correspondence. Adds Tait 1880 bibitem (Proceedings of the Royal Society of Edinburgh, vol. 10). Co-Authored-By: Claude Opus 4.7 --- papers/coloring_nested_tire_graphs/paper.aux | 14 +++-- papers/coloring_nested_tire_graphs/paper.log | 53 ++++++++++--------- papers/coloring_nested_tire_graphs/paper.pdf | Bin 616266 -> 624313 bytes papers/coloring_nested_tire_graphs/paper.tex | 28 ++++++++++ 4 files changed, 64 insertions(+), 31 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 1fdda1a..219ed4f 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -15,13 +15,17 @@ \newlabel{lem:tire-component}{{1.10}{5}} \citation{bauerfeld-pds} \citation{bauerfeld-pds} -\bibcite{bauerfeld-pds}{1} +\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}} +\bibcite{Tait1880}{1} +\bibcite{bauerfeld-pds}{2} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{rem:tire-component-degenerate}{{1.11}{7}} -\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{7}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{7}{}\protected@file@percent } -\gdef \@abspage@last{7} +\@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 d4d81eb..f1b4502 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 18:37 +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 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -213,36 +213,37 @@ File: fig_partial_tire_dual.png Graphic file (type png) Package pdftex.def Info: fig_partial_tire_dual.png used on input line 225. 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z!+n+N`!UJjp#+57f(ORS3M&~r*+^l42m;D^UIwHcp>o#w@%ax1C&a);*O&A;?u|w2 zdEye7pa+KQ+*ZZCr5GjxrQf}=NWIAN(Go;d#G<63smaYta?}!JRK!WhWU7qI@}6Ab zEdGjAL!9tcs>8Z^6stq7POMgi<94k?4dwnU*#!7Jl%)pv@Ruw@z50mLLf5}5Rck@- z48~}%c0ZHWVt7J=>(t&6$h2vGV53eQ6`P+-v%ETFI{qm?Gz0lLXxpy!pAd}J<=9A12!qClV`k^z;D8|~7gv;k F`7c1&DO~^n diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index 1de12bc..1a63966 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -487,8 +487,36 @@ 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). +\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. +\end{remark} + \begin{thebibliography}{9} +\bibitem{Tait1880} +P.~G.~Tait, +\emph{Remarks on the colouring of maps}, +Proceedings of the Royal Society of Edinburgh, vol.~10, pp.~501--503 +and~728--729, 1880. + + \bibitem{bauerfeld-pds} E.~Bauerfeld, \emph{Plane Depth Sequencing},