diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index b6be073..48210bc 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -26,11 +26,10 @@ \newlabel{rem:hamilton-cycle-spoke-only}{{1.13}{9}} \newlabel{rem:bridge-case-theta}{{1.14}{9}} \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}} +\newlabel{rem:count-general-outerplanar}{{1.16}{10}} +\newlabel{thm:tread-tree}{{1.17}{10}} +\newlabel{rem:tree-multiple-children}{{1.18}{11}} \bibcite{tait-original}{1} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tire-duals}{3} @@ -39,7 +38,6 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{rem:tree-multiple-children}{{1.19}{12}} -\newlabel{rem:tree-coloring-factorisation}{{1.20}{12}} +\newlabel{rem:tree-coloring-factorisation}{{1.19}{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 a99f1ec..d2f4a7e 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 02:57 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 03:04 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -513,10 +513,10 @@ LaTeX Warning: `h' float specifier changed to `ht'. [7] [8] [9] [10] [11] [12] (./paper.aux) ) Here is how much of TeX's memory you used: - 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 + 14048 strings out of 478268 + 279229 string characters out of 5846347 + 563840 words of memory out of 5000000 + 31872 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 @@ -546,7 +546,7 @@ fb> -Output written on paper.pdf (12 pages, 623067 bytes). +Output written on paper.pdf (12 pages, 618557 bytes). PDF statistics: 177 PDF objects out of 1000 (max. 8388607) 107 compressed objects within 2 object streams diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index 4a34c08..5045339 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 26d220e..8e1df3b 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -815,66 +815,23 @@ 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. +Theorem~\ref{thm:tait-tire} reduces the $4$-colouring count of a +tire to the $3$-edge-coloring count of its outerplanar inner dual +$\Gamma$. For the cycle case $\Gamma \cong C_n$ (the spoke-only +case of Remark~\ref{rem:hamilton-cycle-spoke-only}), the cycle +chromatic polynomial at $k = 3$ gives +$2^n + 2 (-1)^n$. For an inner dual with one or more non-crossing +chords, 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 number of chords can have different proper +$3$-edge-coloring counts depending on how the chords are arranged +(nested, sequential, sharing vertices, etc.). Every such count +can nevertheless 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]