Counting Group Valued Graph Colorings

TL;DR

This paper develops a Fourier transform-based expansion for counting colorings of graphs with group-valued colors, revealing how group structure influences coloring counts and their asymptotic behavior.

Contribution

It introduces a novel Fourier transform approach to analyze group-valued graph colorings, providing explicit expansions and reciprocity laws that extend classical coloring polynomials.

Findings

Expansion for coloring counts indexed by isthmus free subgraphs

Main term dominated by the empty subgraph when (1-a) is small

Lowest order corrections depend on shortest cycles in the graph

Abstract

There are many variations on partition functions for graph homomorphisms or colorings. The case considered here is a counting or hard constraint problem in which the range or color graph carries a free and vertex transitive Abelian group action so that the colors are identified with the elements of this group. A Fourier transform is used to obtain an expansion for the numbers of colorings with terms indexed by isthmus free subgraphs of the domain. The terms are products of a polynomial in the edge density a of the color graph and the number of colorings of the indexing subgraph of the domain into the complementary color graph. The polynomial in a is independent of the color group and the term has order (1-a) to the r where r is the number of vertices minus the number of components in the indexing subgraph. Thus if (1-a) is small there is a main term indexed by the empty subgraph which is a polynomial in a and the first dependence on the coloring group occurs in the lowest order corrections which are indexed by the shortest cycles in the graph and are of order (1-a) to the g-1 where g is the length of these shortest cycles. The main theorem is stated as a reciprocity law. Examples are given in which the coloring groups are long cycles and products of short cycles and adjacent vertices are required to have distant rather than distinct colors. The chromatic polynomial of a graph corresponds to using any group and taking the allowed set to be the complement of the identity.