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arXiv · 2010.04450

A note on the orientation covering number

Barnabás Janzer
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Abstract

Given a graph $G$, its orientation covering number $\sigma(G)$ is the smallest non-negative integer $k$ with the property that we can choose $k$ orientations of $G$ such that whenever $x, y, z$ are vertices of $G$ with $xy,xz\in E(G)$ then there is a chosen orientation in which both $xy$ and $xz$ are oriented away from $x$. Esperet, Gimbel and King showed that $\sigma(G)\leq \sigma\left(K_{\chi(G)}\right)$, where $\chi(G)$ is the chromatic number of $G$, and asked whether we always have equality. In this note we prove that it is indeed always the case that $\sigma(G)=\sigma(K_{\chi(G)})$. We also determine the exact value of $\sigma(K_n)$ explicitly for `most' values of $n$.

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Barnabás Janzer. 2020-10-09. A note on the orientation covering number. https://doi.org/10.1016/j.dam.2021.08.006

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