diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 1fdda1a..aad7195 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -15,13 +15,15 @@ \newlabel{lem:tire-component}{{1.10}{5}} \citation{bauerfeld-pds} \citation{bauerfeld-pds} +\newlabel{rem:tire-component-degenerate}{{1.11}{7}} +\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{7}} +\newlabel{prop:edge-vertex-bijection}{{1.13}{7}} \bibcite{bauerfeld-pds}{1} \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} +\newlabel{rem:edge-vertex-corollary}{{1.14}{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 ef5d8a5..05af7d5 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 19:48 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 19:52 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -213,36 +213,36 @@ 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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PDF statistics: - 115 PDF objects out of 1000 (max. 8388607) - 67 compressed objects within 1 object stream + 118 PDF objects out of 1000 (max. 8388607) + 69 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 3339df6..c9814c2 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..661c879 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -487,6 +487,66 @@ boundary cycle (the link of $v_0$); the corresponding tire graph has degenerate outer boundary $\{v_0\}$. \end{remark} +\begin{proposition}[Edge--vertex coloring bijection for $D(T)$] +\label{prop:edge-vertex-bijection} +Let $T$ be a tire graph satisfying the spoke-only hypothesis of +Proposition~\ref{prop:partial-tire-dual-structure} (so $D(T) \cong +C_{n+m} \circ K_1$). Let $\Gamma \cong C_{n+m}$ be the interior +dual subgraph of $D(T)$ induced on the interior dual vertices +$\{d_f : f \in F_{\mathrm{ann}}\}$. Then the number of proper +$3$-edge-colorings of $D(T)$ equals the number of proper +$3$-vertex-colorings of $\Gamma$, both given by +\[ + 2^{n+m} + 2 \cdot (-1)^{n+m}. +\] +\end{proposition} + +\begin{proof} +Write $L = n + m$, $\Gamma = C_L$. We construct mutually inverse +bijections. + +\emph{Step 1: proper $3$-edge-colorings of $D(T)$ $\leftrightarrow$ +proper $3$-edge-colorings of $C_L$.} Given a proper $3$-edge-coloring +$\chi$ of $D(T)$, the three edges incident to any $d_f$ carry three +distinct colors; in particular the two cycle edges incident to $d_f$ +carry distinct colors, so $\chi|_{E(C_L)}$ is a proper $3$-edge-coloring +of $C_L$. Conversely, given a proper $3$-edge-coloring $\psi$ of +$C_L$, the two cycle edges at any $d_f$ have distinct colors, so a +unique third color is available; assign that color to $d_f$'s leaf +edge. The resulting extension to $D(T)$ is proper at every $d_f$ and +vacuously proper at every leaf (degree~$1$), and the two maps are +inverse to each other. Therefore +\[ + \#\bigl\{\text{proper $3$-edge-colorings of } D(T)\bigr\} + \;=\; + \#\bigl\{\text{proper $3$-edge-colorings of } C_L\bigr\}. +\] + +\emph{Step 2: proper $3$-edge-colorings of $C_L$ $\leftrightarrow$ +proper $3$-vertex-colorings of $L(C_L) \cong C_L$.} The line graph +$L(C_L)$ of a cycle of length $L$ is again a cycle of length $L$; +proper edge-colorings of $C_L$ are by definition proper vertex-colorings +of $L(C_L)$. + +\emph{Step 3: count.} The chromatic polynomial of the cycle is +$P(C_L, k) = (k-1)^L + (-1)^L (k-1)$; at $k = 3$ this gives +$2^L + 2 \cdot (-1)^L$. +\end{proof} + +\begin{remark} +\label{rem:edge-vertex-corollary} +Proposition~\ref{prop:edge-vertex-bijection} reduces counting proper +$3$-edge-colorings of $D(T)$ to counting proper $3$-vertex-colorings of +a single cycle, giving a closed form $2^{n+m} + 2(-1)^{n+m}$ that +depends only on $n+m$ (not on the specific spoke-only annular +triangulation, nor on the chord structure of $O$). The count is +preserved under the corona-with-$K_1$ structure of +Proposition~\ref{prop:partial-tire-dual-structure} precisely because +each degree-$1$ leaf imposes no proper-edge-coloring constraint on +itself; its color is freely determined as the missing third color at +its attached interior vertex. +\end{remark} + \begin{thebibliography}{9} \bibitem{bauerfeld-pds}