diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 7329362..b6be073 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -25,15 +25,21 @@ \newlabel{fig:inner-dual-annulus-case}{{4}{9}} \newlabel{rem:hamilton-cycle-spoke-only}{{1.13}{9}} \newlabel{rem:bridge-case-theta}{{1.14}{9}} -\newlabel{thm:tread-tree}{{1.15}{10}} -\bibcite{bauerfeld-depth}{1} -\bibcite{bauerfeld-nested-tire-duals}{2} +\citation{tait-original} +\citation{bauerfeld-nested-tire-duals} +\newlabel{thm:tait-tire}{{1.15}{10}} +\newlabel{thm:count-formula}{{1.16}{10}} +\newlabel{rem:count-general-outerplanar}{{1.17}{11}} +\newlabel{thm:tread-tree}{{1.18}{11}} +\bibcite{tait-original}{1} +\bibcite{bauerfeld-depth}{2} +\bibcite{bauerfeld-nested-tire-duals}{3} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{rem:tree-multiple-children}{{1.16}{11}} -\newlabel{rem:tree-coloring-factorisation}{{1.17}{11}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{11}{}\protected@file@percent } -\gdef \@abspage@last{11} +\newlabel{rem:tree-multiple-children}{{1.19}{12}} +\newlabel{rem:tree-coloring-factorisation}{{1.20}{12}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{12}{}\protected@file@percent } +\gdef \@abspage@last{12} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index f23232a..a99f1ec 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) 27 MAY 2026 02:40 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 02:57 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -511,47 +511,45 @@ Package pdftex.def Info: fig_tire_example.png used on input line 179. LaTeX Warning: `h' float specifier changed to `ht'. -[7] [8] [9] -Underfull \vbox (badness 10000) has occurred while \output is active [] - - [10] -[11] (./paper.aux) ) +[7] [8] [9] [10] [11] [12] (./paper.aux) ) Here is how much of TeX's memory you used: - 14043 strings out of 478268 - 279149 string characters out of 5846347 - 563813 words of memory out of 5000000 - 31868 multiletter control sequences out of 15000+600000 - 477909 words of font info for 61 fonts, out of 8000000 for 9000 + 14049 strings out of 478268 + 279248 string characters out of 5846347 + 563852 words of memory out of 5000000 + 31873 multiletter control sequences out of 15000+600000 + 478218 words of font info for 62 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 84i,12n,89p,1156b,803s stack positions out of 10000i,1000n,20000p,200000b,200000s -< -/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb> -Output written on paper.pdf (11 pages, 603044 bytes). + + +Output written on paper.pdf (12 pages, 623067 bytes). PDF statistics: - 169 PDF objects out of 1000 (max. 8388607) - 102 compressed objects within 2 object streams + 177 PDF objects out of 1000 (max. 8388607) + 107 compressed objects within 2 object streams 0 named destinations out of 1000 (max. 500000) 23 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 5882ceb..4a34c08 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 f8b40f7..26d220e 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -757,6 +757,126 @@ and so contributes no degree-$2$ branch vertex), hence is outerplanar as predicted. \end{remark} +\begin{theorem}[Tait correspondence: 4-colorings of a tire vs 3-edge-colorings of its inner dual] +\label{thm:tait-tire} +Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph +(viewed as an annular triangulation of its tire tread $R$) and let +$\Gamma$ be its inner dual +(Theorem~\ref{thm:inner-dual-outerplanar}). Then +\[ + \#\bigl\{\text{proper $4$-vertex-colorings of } T\bigr\} \big/ |S_4| + \;=\; + \#\bigl\{\text{proper $3$-edge-colorings of } \Gamma\bigr\} \big/ |S_3|. +\] +That is, the number of $4$-vertex-colorings of $T$ up to permutation +of the colour set $\{0, 1, 2, 3\}$ equals the number of +$3$-edge-colorings of $\Gamma$ up to permutation of the colour set +$\{1, 2, 3\}$. +\end{theorem} + +\begin{proof} +The argument is the classical Tait correspondence +\cite{tait-original} adapted to the annular triangulation $T$. +Encode the four colours of a proper $4$-vertex-coloring $c \colon +V(T) \to \mathbb{Z}_2 \times \mathbb{Z}_2$. For each interior +annular edge $e$ of $T$ (whose two incident faces both lie in +$F_{\mathrm{ann}}$, contributing a $\Gamma$-edge $e^*$), set +\[ + \chi^*(e^*) \;:=\; c(u) + c(v) \in \mathbb{Z}_2 \times \mathbb{Z}_2, + \qquad \text{where } u, v \text{ are the endpoints of } e. +\] +Since $c(u) \ne c(v)$, we have $\chi^*(e^*) \ne 00$, so $\chi^*$ +takes values in $\{01, 10, 11\}$, which we identify with the +$3$-edge-coloring palette $\{1, 2, 3\}$. + +\emph{Properness.} At each $\Gamma$-vertex $d_f$ corresponding to +an annular triangle $f = \{u, v, w\}$, the three incident +$\Gamma$-edges (one per cycle-edge of $f$) carry colours +$c(u) + c(v),\; c(v) + c(w),\; c(u) + c(w)$. These three elements +of $\mathbb{Z}_2 \times \mathbb{Z}_2$ sum to $0$ and are pairwise +distinct (their pairwise differences are +$c(u) - c(w),\; c(v) - c(u),\; c(w) - c(v)$, each nonzero), so +they form a permutation of $\{01, 10, 11\}$ --- a proper edge +colouring at $d_f$. + +\emph{Surjectivity onto cosets.} Given a proper $3$-edge-coloring +$\chi^*$ of $\Gamma$, the equation $c(u) + c(v) = \chi^*(e^*)$ +admits exactly $|\mathbb{Z}_2 \times \mathbb{Z}_2| = 4$ solutions +$c \colon V(T) \to \mathbb{Z}_2 \times \mathbb{Z}_2$ (a global +translation is the only freedom). Hence the map $c \mapsto +\chi^*$ is $4$-to-$1$. + +\emph{Count.} Therefore $\#\{\text{4-colorings of } T\} = 4 \cdot +\#\{\text{3-edge-colorings of } \Gamma\}$. Dividing by $|S_4| += 24$ on the left and $|S_3| = 6$ on the right (since $S_4$ acts +faithfully on the $4$-colorings and $S_3$ on the $3$-edge-colorings, +and the $4$-to-$1$ map respects the $S_4/S_3 \cong S_3$ quotient +via the natural surjection $S_4 \twoheadrightarrow S_3$) gives the +stated equality. +\end{proof} + +\begin{theorem}[Count formula for spoke-only and single-chord tires] +\label{thm:count-formula} +Let $T$ be a tire graph with $|F_{\mathrm{ann}}| = n$ annular +triangles, and let $\Gamma$ be its inner dual. +\begin{enumerate} +\item[(i)] If $\Gamma \cong C_n$ (the spoke-only case + of Remark~\ref{rem:hamilton-cycle-spoke-only}), then + \[ + \#\bigl\{\text{proper $3$-edge-colorings of } \Gamma\bigr\} + \;=\; 2^n + 2 \cdot (-1)^n. + \] +\item[(ii)] If $\Gamma \cong \Theta(1, b, c)$ with $b + c = n$ + (the single-chord case of + Remark~\ref{rem:bridge-case-theta}), then + \[ + \#\bigl\{\text{proper $3$-edge-colorings of } \Gamma\bigr\} + \;=\; 6 \,(\alpha_b \alpha_c + \beta_b \beta_c), + \] + where + $\alpha_L = \bigl(2^{L-1} + 2 (-1)^{L-1}\bigr) / 3$ and + $\beta_L = \bigl(2^{L-1} - (-1)^{L-1}\bigr) / 3$. +\end{enumerate} +\end{theorem} + +\begin{proof} +\emph{(i)} Standard chromatic polynomial of the cycle: $P(C_n, k) += (k - 1)^n + (-1)^n (k - 1)$. At $k = 3$: +$2^n + 2 (-1)^n$ (cf.\ Proposition~1.2 of +\cite{bauerfeld-nested-tire-duals}). + +\emph{(ii)} By transfer matrix on the two non-chord paths of +$\Theta(1, b, c)$ with the chord edge colour fixed and the two +endpoint colour assignments enumerated. At each trivalent endpoint, +the two path-incident edges receive the two non-chord colours +($2$ ways). Conditional on these assignments, the path's interior +edges form a $3$-edge-coloring of a path of length $b$ (resp.\ $c$) +with both endpoints' colours fixed; the number of such colorings is +the $(c_a, c_b)$-entry of $T^{L-1}$, where $T = J - I$ is the +$3 \times 3$ adjacency matrix of the colour-difference graph. +$T^{L-1}$ has diagonal entries $\alpha_L$ and off-diagonal entries +$\beta_L$. Summing over the four endpoint configurations +($\{x, y\}, \{x', y'\} = \{2, 3\}$ each in two orderings) and +multiplying by the three chord colour choices gives the stated +formula. Verification: $\Theta(1, 2, 2)$ (= $K_4 \setminus e$, +$\alpha_2 = 0,\; \beta_2 = 1$) yields $6 \cdot (0 + 1) = 6$ proper +$3$-edge-colorings, matching the known count for $K_4 \setminus e$. +\end{proof} + +\begin{remark} +\label{rem:count-general-outerplanar} +For an inner dual $\Gamma$ with more than one non-crossing chord, +the count depends on the chord structure, not just on the pair +(number of vertices, number of chords). Two outerplanar graphs +with the same $n$ and $k$ can have different proper $3$-edge-coloring +counts depending on how the chords are arranged (nested, +sequential, sharing vertices, etc.). However, every such count +can be computed in linear time by tree-decomposition methods, +since outerplanar graphs have treewidth at most $2$ and the +edge-chromatic polynomial admits a deletion--contraction recursion +that respects the cycle-plus-chord structure. +\end{remark} + \begin{theorem}[Tire treads form a rooted tree under face containment] \label{thm:tread-tree} Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ @@ -876,6 +996,11 @@ program for tire treads. \begin{thebibliography}{9} +\bibitem{tait-original} +P.~G.~Tait, +\emph{Remarks on the colouring of maps}, +Proc.\ Roy.\ Soc.\ Edinburgh \textbf{10} (1880), 729. + \bibitem{bauerfeld-depth} E.~Bauerfeld, \emph{Plane Depth},