Counting Group Valued Graph Colorings
Eric Babson, Matthias Beck

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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Graph Theory Research · Advanced Combinatorial Mathematics · Limits and Structures in Graph Theory
