Graph homomorphisms via vector colorings

Abstract

In this paper we study the existence of homomorphisms G→H using semidefinite programming. Specifically, we use the vector chromatic number of a graph, defined as the smallest real number t≥2 for which there exists an assignment of unit vectors i↦pi to its vertices such that 〈pi,pj〉≤−1∕(t−1), when i∼j. Our approach allows to reprove, without using the Erdős–Ko–Rado Theorem, that for n>2r the Kneser graph Kn:r and the q-Kneser graph qKn:r are cores, and furthermore, that for n∕r=n′∕r′ there exists a homomorphism Kn:r→Kn′:r′ if and only if n divides n′. In terms of new applications, we show that the even-weight component of the distance k-graph of the n-cube Hn,k is a core and also, that non-bipartite Taylor graphs are cores. Additionally, we give a necessary and sufficient condition for the existence of homomorphisms Hn,k→Hn′,k′ when n∕k=n′∕k′. Lastly, we show that if a 2-walk-regular graph (which is non-bipartite and not complete multipartite) has a unique optimal vector coloring, it is a core. Based on this sufficient condition we conducted a computational study on Ted Spence’s list of strongly regular graphs (http://www.maths.gla.ac.uk/ es/srgraphs.php) and found that at least 84% are cores.