Chromatic functors of graphs

TL;DR

This paper introduces a functorial approach to graph coloring, showing that finite graphs with the same chromatic polynomial have isomorphic chromatic functors, and explores extensions to infinite graphs.

Contribution

It formalizes chromatic polynomials as functors, establishing a categorical framework for graph coloring and extending it to infinite graphs with new properties.

Findings

Finite graphs with same chromatic polynomial have isomorphic chromatic functors

Chromatic functors satisfy the Cantor-Bernstein-Schr"oder property

Countable connected trees have isomorphic chromatic functors

Abstract

Finite graphs that have a common chromatic polynomial have the same number of regular $n$-colorings. A natural question is whether there exists a natural bijection between regular $n$-colorings. We address this question using a functorial formulation. Let $G$ be a simple graph. Then for each set $X$ we can associate a set of $X$-colorings. This defines a functor, "chromatic functor" from the category of sets with injections to itself. The first main result verifies that two finite graphs determine isomorphic chromatic functors if and only if they have the same chromatic polynomial. Chromatic functors can be defined for arbitrary, possibly infinite, graphs. This fact enables us to investigate functorial chromatic theory for infinite graphs. We prove that chromatic functors satisfy the Cantor-Bernstein-Schr\"oder property. We also prove that countable connected trees determine isomorphic chromatic functors. Finally, we present a pair of infinite graphs that determine non-isomorphic chromatic functors.