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

TL;DR

This work develops an $\\alpha$-representation for counting Tait colorings of a maximal planar graph by translating edge colorings into Gaussian-weighted sums over spanning-tree–related contractions in the finite field $\\mathbb{F}_3$. The method combines a Heawood-type representation of Tai$_0(G)$, a Fourier transform over $\\mathbb{F}_3$, and a Gaussian-sum analysis of the Laplace–Kirchhoff matrix to express Tai$_0(G)$ as a tripled sum over face spins $\\alpha$ with edge weights $x_e(\\alpha)=\\alpha(F'_e)+\\alpha(F''_e)$. The main result shows Tai$_0(G)$ equals a sum of weights $w_G(x(\\alpha))$ (or equivalently Gaussian sums of $L(x( -\\alpha))$) over all admissible $\\alpha$, with independence from the contraction set $W^*(x)$. This provides a new algebraic framework linking Tait colorings, spanning-tree sums, and Gaussian sums, with implications tied to the Four Color Theorem and potential extensions to representable matroids.

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 $α(F)=\pm 1 \in \mathbb{F}_3$. Then we can associate each edge with $x_e=α(F'_e)+α(F''_e)$, 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 $w_G({\mathbf x}(α))$ with respect to all possible vectors $α\in \{-1, 1\}^{\mathcal F(G)}$ such that $G/W^*({\mathbf x}(α))$ 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.

Figures (1)

• Figure 1: Three cases of vector $\alpha$ for $K_4$

Theorems & Definitions (11)

• Theorem 1
• Remark 1
• Lemma 1: a particular case of Lemma 8 in EJC for the field $\mathbb F_3$
• Corollary 1
• Corollary 2
• proof
• Lemma 2: cf. Theorem 6 in EJC
• proof
• Proposition 1: heawood
• Remark 2