Graph homomorphisms via vector colorings - Godsil, Chris (2020)
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.