On Orthogonal Vector Edge Coloring

A Silva, A Ibiapina - arXiv preprint arXiv:1909.01918, 2019 - arxiv.org
A Silva, A Ibiapina
arXiv preprint arXiv:1909.01918, 2019arxiv.org
Given a graph $ G $ and a positive integer $ d $, an orthogonal vector $ d $-coloring of $ G $
is an assignment $ f $ of vectors of $\mathbb {R}^ d $ to $ V (G) $ in such a way that adjacent
vertices receive orthogonal vectors. The orthogonal chromatic number of $ G $, denoted by
$\chi_v (G) $, is the minimum $ d $ for which $ G $ admits an orthogonal vector $ d $-
coloring. This notion has close ties with the notions of Lov\'asz Theta Function, quantum
chromatic number, and many other problems, and even though this and related metrics have …
Given a graph and a positive integer , an orthogonal vector -coloring of is an assignment of vectors of to in such a way that adjacent vertices receive orthogonal vectors. The orthogonal chromatic number of , denoted by , is the minimum for which admits an orthogonal vector -coloring. This notion has close ties with the notions of Lov\'asz Theta Function, quantum chromatic number, and many other problems, and even though this and related metrics have been extensively studied over the years, we have found that there is a gap in the knowledge concerning the edge version of the problem. In this article, we discuss this version and its relation with other insteresting known facts, and pose a question about the orthogonal chromatic index of cubic graphs.
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