From d121d2d3b68de02faf2202e4456de7f0db6aab1a Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 19:52:31 -0400 Subject: [PATCH] coloring_nested_tire_graphs: prove edge-vertex coloring bijection for D(T) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Adds Proposition 1.13 (Edge-vertex coloring bijection for D(T)): for a tire graph T satisfying the spoke-only hypothesis of Prop 1.8 (so D(T) ~= C_{n+m} ∘ K_1), the number of proper 3-edge-colorings of D(T) equals the number of proper 3-vertex-colorings of its interior dual subgraph Γ ~= C_{n+m}, and both equal 2^{n+m} + 2 · (-1)^{n+m}. Proof: Two bijection steps. Step 1: Restriction is a bijection between proper 3-edge-colorings of D(T) and proper 3-edge-colorings of the cycle C_L (where L = n+m), because at each d_f the leaf's color is forced to be the unique third color absent from the two cycle edges, and leaves impose no further constraint. Step 2: Proper 3-edge-colorings of C_L = proper 3-vertex-colorings of L(C_L) = proper 3-vertex-colorings of C_L (since L(C_L) ~= C_L). Step 3: Chromatic polynomial of C_L at k=3 is 2^L + 2 · (-1)^L. Adds Remark 1.14 noting the closed form depends only on n+m, not on the specific spoke-only annular triangulation or chord structure of O. Empirically verified for L in [3, 10] via Sage's chromatic polynomials: edge-3-colorings of D(T) = vertex-3-colorings of C_L = formula in every case. Paper grows from 7 to 8 pages. Co-Authored-By: Claude Opus 4.7 --- papers/coloring_nested_tire_graphs/paper.aux | 10 ++-- papers/coloring_nested_tire_graphs/paper.log | 52 ++++++++-------- papers/coloring_nested_tire_graphs/paper.pdf | Bin 616266 -> 622907 bytes papers/coloring_nested_tire_graphs/paper.tex | 60 +++++++++++++++++++ 4 files changed, 92 insertions(+), 30 deletions(-) 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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+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}