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coloring_nested_tire_graphs: state Tait correspondence on partial tire dual; cite Tait 1880

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Adds Proposition 1.13: the number of non-equivalent proper 4-vertex-
colorings of a tire graph T (mod S_4) equals the number of non-
equivalent proper 3-edge-colorings of its partial tire dual D(T) (mod
S_3).  The map is the classical Tait XOR construction: identifying
the four colors with Z_2 x Z_2, each edge of T receives an edge color
equal to the XOR of its endpoint colors, which lies in the three
nonzero elements of Z_2 x Z_2 -- giving the corresponding edge of
D(T) a 3-edge-color.  Annular triangles of T, encoded as degree-3
vertices d_f of D(T), supply the three-distinct-colors constraint.

Adds Remark 1.14 explaining the analogy with Tait's classical
correspondence.

Adds Tait 1880 bibitem (Proceedings of the Royal Society of Edinburgh,
vol. 10).

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-25 18:48:10 -04:00
parent e212ada649
commit 35218f2191
4 changed files with 64 additions and 31 deletions
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papers/coloring_nested_tire_graphs/paper.aux
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\newlabel{lem:tire-component}{{1.10}{5}} \newlabel{lem:tire-component}{{1.10}{5}}
\citation{bauerfeld-pds} \citation{bauerfeld-pds}
\citation{bauerfeld-pds} \citation{bauerfeld-pds}
\bibcite{bauerfeld-pds}{1} \citation{Tait1880}
\newlabel{rem:tire-component-degenerate}{{1.11}{7}}
\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{7}}
\newlabel{prop:tait-tire}{{1.13}{7}}
\newlabel{rem:tait}{{1.14}{7}}
\bibcite{Tait1880}{1}
\bibcite{bauerfeld-pds}{2}
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papers/coloring_nested_tire_graphs/paper.log
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papers/coloring_nested_tire_graphs/paper.tex
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@@ -487,8 +487,36 @@ 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}[Tait correspondence on the partial tire dual]
\label{prop:tait-tire}
The number of non-equivalent proper $4$-vertex-colorings of a tire
graph $T$ (modulo permutation of the four colors) equals the number
of non-equivalent proper $3$-edge-colorings of its partial tire dual
$D(T)$ (modulo permutation of the three colors).
\end{proposition}
\begin{remark}
\label{rem:tait}
Proposition~\ref{prop:tait-tire} is the tire-graph analogue of Tait's
classical correspondence~\cite{Tait1880}: identifying the four colors
with the elements of $\mathbb{Z}_2 \times \mathbb{Z}_2$, the XOR of
the two endpoint colors of an edge of $T$ lies in the three nonzero
elements of $\mathbb{Z}_2 \times \mathbb{Z}_2$ and assigns a proper
$3$-edge-coloring to the corresponding edge of $D(T)$. The annular
triangles of $T$, encoded as the degree-$3$ vertices $d_f$ of $D(T)$,
contribute the requirement that each $d_f$'s three incident edges
carry three distinct colors.
\end{remark}
\begin{thebibliography}{9} \begin{thebibliography}{9}
\bibitem{Tait1880}
P.~G.~Tait,
\emph{Remarks on the colouring of maps},
Proceedings of the Royal Society of Edinburgh, vol.~10, pp.~501--503
and~728--729, 1880.
\bibitem{bauerfeld-pds} \bibitem{bauerfeld-pds}
E.~Bauerfeld, E.~Bauerfeld,
\emph{Plane Depth Sequencing}, \emph{Plane Depth Sequencing},