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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
Software:
SageMath
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