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. (pdftex.def) Requested size: 280.79956pt x 233.36552pt. - [4 <./fig_partial_tire_dual.png>] [5] [6] [7] (./paper.aux) ) + [4 <./fig_partial_tire_dual.png>] [5] [6] [7] [8] (./paper.aux) ) Here is how much of TeX's memory you used: - 3018 strings out of 478268 - 42332 string characters out of 5846347 - 345207 words of memory out of 5000000 - 21064 multiletter control sequences out of 15000+600000 + 3021 strings out of 478268 + 42368 string characters out of 5846347 + 345232 words of memory out of 5000000 + 21067 multiletter control sequences out of 15000+600000 475666 words of font info for 53 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 69i,8n,76p,687b,316s stack positions out of 10000i,1000n,20000p,200000b,200000s - -Output written on paper.pdf (7 pages, 616266 bytes). + +Output written on paper.pdf (8 pages, 624313 bytes). PDF statistics: - 115 PDF objects out of 1000 (max. 8388607) - 67 compressed objects within 1 object stream + 123 PDF objects out of 1000 (max. 8388607) + 72 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 16 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index d1e2ddb..6f0b268 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 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},