The $α$-representation for Tait coloring and sums over spanning trees
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| Formato: | Preprint |
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2025
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| _version_ | 1866915546683932672 |
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| author | Kalimullin, Ilyas Lerner, Eduard |
| author_facet | Kalimullin, Ilyas Lerner, Eduard |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10213 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $α$-representation for Tait coloring and sums over spanning trees Kalimullin, Ilyas Lerner, Eduard Combinatorics Number Theory 05C15 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. |
| title | The $α$-representation for Tait coloring and sums over spanning trees |
| topic | Combinatorics Number Theory 05C15 |
| url | https://arxiv.org/abs/2510.10213 |