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More Tales of Hoffman: bounds for the vector chromatic number of a graph

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arxiv 1812.02613 v3 pith:KK2MGNZ5 submitted 2018-12-06 math.CO

Pawel Wocjan , Clive Elphick , David Anekstein This is my paper · ORCID
classification math.CO
keywords boundslowerchromaticnumberbounddenotegraphgraphs
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abstract

Let $\chi(G)$ denote the chromatic number of a graph and $\chi_v(G)$ denote the vector chromatic number. For all graphs $\chi_v(G) \le \chi(G)$ and for some graphs $\chi_v(G) \ll \chi(G)$. Galtman proved that Hoffman's well-known lower bound for $\chi(G)$ is in fact a lower bound for $\chi_v(G)$. We prove that two more spectral lower bounds for $\chi(G)$ are also lower bounds for $\chi_v(G)$. We then use one of these bounds to derive a new characterization of $\chi_v(G)$.

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  1. Conic programming to understand sums of squares of eigenvalues of graphs

    math.CO 2024-11 conditional novelty 7.0 of 10

    For every graph, min{s+, s-} is at least 2m/chi_vec(G), resolving a conjecture of Wocjan, Elphick and Anekstein.

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