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Graph homomorphisms via vector colorings. (English) Zbl 1414.05199
Summary: In this paper we study the existence of homomorphisms $$G \to H$$ using semidefinite programming. Specifically, we use the vector chromatic number of a graph, defined as the smallest real number $$t \geq 2$$ for which there exists an assignment of unit vectors $$i \mapsto p_i$$ to its vertices such that $$\langle p_i, p_j \rangle \leq - 1 /(t - 1),$$ when $$i \sim j$$. Our approach allows to reprove, without using the Erdős-Ko-Rado theorem, that for $$n > 2 r$$ the Kneser graph $$K_{n : r}$$ and the $$q$$-Kneser graph $$q K_{n : r}$$ are cores, and furthermore, that for $$n / r = n^\prime / r^\prime$$ there exists a homomorphism $$K_{n : r} \to K_{n^\prime : r^\prime}$$ if and only if $$n$$ divides $$n^\prime$$. 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 $$H_{n, k} \to H_{n^\prime, k^\prime}$$ when $$n / k = n^\prime /k^\prime$$. 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 [“Strongly regular graphs on at most 64 vertices”, http://www.maths.gla.ac.uk/~es/srgraphs.php] and found that at least 84% are cores.

MSC:
 05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) 05C15 Coloring of graphs and hypergraphs 90C22 Semidefinite programming
Keywords:
Kneser graph; Taylor graphs
SageMath
Full Text:
References:
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