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coloring_nested_tire_graphs: prove edge-vertex coloring bijection for D(T)

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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 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-25 19:52:31 -04:00
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\newlabel{lem:tire-component}{{1.10}{5}} \newlabel{lem:tire-component}{{1.10}{5}}
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@@ -487,6 +487,66 @@ boundary cycle (the link of $v_0$); the corresponding tire graph has
degenerate outer boundary $\{v_0\}$. degenerate outer boundary $\{v_0\}$.
\end{remark} \end{remark}
\begin{proposition}[Edge--vertex coloring bijection for $D(T)$]
\label{prop:edge-vertex-bijection}
Let $T$ be a tire graph satisfying the spoke-only hypothesis of
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} \begin{thebibliography}{9}
\bibitem{bauerfeld-pds} \bibitem{bauerfeld-pds}