coloring_nested_tire_graphs: drop count-formula theorem, keep remark

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>

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}{{1.16}{10}}
-\newlabel{rem:count-general-outerplanar}{{1.17}{11}}
+\newlabel{thm:tread-tree}{{1.17}{10}}
-\newlabel{thm:tread-tree}{{1.18}{11}}
+\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}{{1.19}{12}}
-\newlabel{rem:tree-coloring-factorisation}{{1.20}{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

@@ -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,
+Theorem~\ref{thm:tait-tire} reduces the $4$-colouring count of a
-the count depends on the chord structure, not just on the pair
+tire to the $3$-edge-coloring count of its outerplanar inner dual
-(number of vertices, number of chords).  Two outerplanar graphs
+$\Gamma$.  For the cycle case $\Gamma \cong C_n$ (the spoke-only
-with the same $n$ and $k$ can have different proper $3$-edge-coloring
+case of Remark~\ref{rem:hamilton-cycle-spoke-only}), the cycle
-counts depending on how the chords are arranged (nested,
+chromatic polynomial at $k = 3$ gives
-sequential, sharing vertices, etc.).  However, every such count
+$2^n + 2 (-1)^n$.  For an inner dual with one or more non-crossing
-can be computed in linear time by tree-decomposition methods,
+chords, the count depends on the chord structure, not just on the
-since outerplanar graphs have treewidth at most $2$ and the
+pair (number of vertices, number of chords): two outerplanar graphs
-edge-chromatic polynomial admits a deletion--contraction recursion
+with the same $n$ and number of chords can have different proper
-that respects the cycle-plus-chord structure.
+$3$-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]