Graph homomorphisms via vector colorings

Chris Godsil, David E. Roberson, Brendan Rooney, Robert Šámal, Antonios Varvitsiotis

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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↦p i to its vertices such that 〈p i ,p j 〉≤−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 K n:r and the q-Kneser graph qK n:r are cores, and furthermore, that for n∕r=n ∕r there exists a homomorphism K n:r →K n :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 H n,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 H n,k →H n ,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.

Original languageEnglish
JournalEuropean Journal of Combinatorics
Volume79
Pages (from-to)244-261
ISSN0195-6698
DOIs
Publication statusPublished - 1 Jun 2019

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