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coloring_nested_tire_graphs: drop count-formula theorem, keep remark

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Removed Theorem 1.16 (the count formula for spoke-only and
single-chord cases). Folded the cycle formula 2^n + 2(-1)^n into
the surviving remark so the only retained content is the structural
observation:

  - Tait reduces 4-coloring count to 3-edge-coloring count of Γ.
  - For Γ ≅ C_n (spoke-only): cycle chromatic polynomial gives
    2^n + 2(-1)^n.
  - For Γ with chords, the count depends on chord structure
    (nested vs. sequential etc.), not just (n, k).
  - Always computable in linear time via tree decomposition
    (outerplanar has treewidth ≤ 2).

Page count: 12 → 11.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-27 03:04:31 -04:00
parent 7801ce959e
commit 787d9f0caf
4 changed files with 25 additions and 70 deletions
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papers/coloring_nested_tire_graphs/paper.aux
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@@ -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}
papers/coloring_nested_tire_graphs/paper.log
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papers/coloring_nested_tire_graphs/paper.tex
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@@ -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]