Graph Homomorphisms via Vector Colorings
Chris Godsil, David E. Roberson, Brendan Rooney, Robert, \v{S}\'amal, Antonios Varvitsiotis

TL;DR
This paper introduces a semidefinite programming approach using vector colorings to analyze graph homomorphisms and cores, providing new proofs and characterizations for various classes of graphs.
Contribution
It presents a novel method to study graph homomorphisms via vector colorings, offering new proofs, characterizations, and applications for specific graph families.
Findings
Kneser and q-Kneser graphs are cores for n>2r.
Existence of homomorphisms depends on divisibility conditions.
Majority of studied strongly regular graphs are cores.
Abstract
In this paper we study the existence of homomorphisms using semidefinite programming. Specifically, we use the vector chromatic number of a graph, defined as the smallest real number for which there exists an assignment of unit vectors to its vertices such that when . Our approach allows to reprove, without using the Erd\H{o}s-Ko-Rado Theorem, that for the Kneser graph and the -Kneser graph are cores, and furthermore, that for there exists a homomorphism if and only if divides . In terms of new applications, we show that the even-weight component of the distance -graph of the -cube is a core and also, that non-bipartite Taylor graphs are cores. Additionally, we give a necessary and sufficient condition for the…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
