Vector Coloring the Categorical Product of Graphs

TL;DR

This paper proves that the vector chromatic number of the categorical product of two graphs equals the minimum of their individual vector chromatic numbers, extending the Hedetniemi Conjecture to vector colorings and exploring related semidefinite programming tools.

Contribution

It establishes the equality of the vector chromatic number for graph products, providing a vector coloring analog of Hedetniemi's Conjecture and characterizing when optimal colorings are composable.

Findings

Proves vector chromatic number of G×H equals min of those of G and H.

Provides necessary and sufficient conditions for combining optimal colorings.

Connects vector chromatic number to Lovász theta function and semidefinite programming.

Abstract

A vector $t$-coloring of a graph is an assignment of real vectors $p_1, \ldots, p_n$ to its vertices such that $p_i^Tp_i = t-1$ for all $i=1, \ldots, n$ and $p_i^Tp_j \le -1$ whenever $i$ and $j$ are adjacent. The vector chromatic number of $G$ is the smallest real number $t \ge 1$ for which a vector $t$-coloring of $G$ exists. For a graph $H$ and a vector $t$-coloring $p_1,\ldots,p_n$ of a graph $G$, the assignment $(i,\ell) \mapsto p_i$ is a vector $t$-coloring of the categorical product $G \times H$. It follows that the vector chromatic number of $G \times H$ is at most the minimum of the vector chromatic numbers of the factors. We prove that equality always holds, constituting a vector coloring analog of the famous Hedetniemi Conjecture from graph coloring. Furthermore, we prove a necessary and sufficient condition for when all of the optimal vector colorings of the product can be expressed in terms of the optimal vector colorings of the factors. The vector chromatic number is closely related to the well-known Lov\'{a}sz theta function, and both of these parameters admit formulations as semidefinite programs. This connection to semidefinite programming is crucial to our work and the tools and techniques we develop could likely be of interest to others in this field.