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 bar right arrow p(i) to its vertices such that < p(i), p(j)> <= -1/(t - 1), when i similar to j. Our approach allows to reprove, without using the Erdos-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,H-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',H-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.a c.uk/similar to es/srgraphs.php) and found that at least 84% are cores. (C) 2019 Elsevier Ltd. All rights reserved.