The $\alpha$-representation for Tait coloring and sums over spanning trees

Abstract

Consider a connected pseudograph $H$ such that each edge is associated with weight $x_e$, $x_e \in \mathbb{F}_3$; $\mathcal{T}(H)$ is the set of spanning trees of graph $H$. Assume that $s(H;{\mathbf x})=\sum_{T\in\mathcal{T}(H)} \prod_{e\in E(T)} x_e$. Let $G$ be a maximal planar graph (arbitrary planar triangulation) such that each face $F$ is assigned the value $\alpha(F)=\pm 1 \in \mathbb{F}_3$. Then we can associate each edge with $x_e=\alpha(F'_e)+\alpha(F''_e)$, where $F'_e$ and $F''_e$ are the faces containing edge $e$. Let us define the value $w_G({\mathbf x})$ as $\left(\frac{s(G/W^*({\mathbf x});{\mathbf x})}3\right)/(-3)^{\left(|V(G/W^*({\mathbf x}))| - 1\right)/2}$; here $\left(\frac{x}3\right)$ is the Legendre symbol, $G/W$ is the graph with the contracted set of vertices $W$, while $W^*({\mathbf x})$ is a set of vertices $W$, $W \subseteq V(G)$, with minimal cardinality such that $s(G/W;{\mathbf x})$ differs from zero. In the following, we prove that the number of Tait colorings for graph $G$ equals the tripled sum $w_G({\mathbf x}(\alpha))$ with respect to all possible vectors $\alpha \in \{-1, 1\}^{\mathcal F(G)}$ such that $G/W^*({\mathbf x}(\alpha))$ has an odd number of vertices, where $\mathcal F(G)$ is the set of faces of graph $G$. Keywords: maximal planar graph, Tait coloring, Laplace-Kirchhoff matrix, spanning tree.