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arXiv · 2101.04768

Strong edge colorings of graphs and the covers of Kneser graphs

Borut Lužar·Edita Mačajová·Martin Škoviera·Roman Soták
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Abstract

A proper edge coloring of a graph is strong if it creates no bichromatic path of length three. It is well known that for a strong edge coloring of a $k$-regular graph at least $2k-1$ colors are needed. We show that a $k$-regular graph admits a strong edge coloring with $2k-1$ colors if and only if it covers the Kneser graph $K(2k-1,k-1)$. In particular, a cubic graph is strongly $5$-edge-colorable whenever it covers the Petersen graph. One of the implications of this result is that a conjecture about strong edge colorings of subcubic graphs proposed by Faudree et al. [Ars Combin. 29 B (1990), 205--211] is false.

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Borut Lužar, Edita Mačajová, Martin Škoviera, Roman Soták. 2021-01-12. Strong edge colorings of graphs and the covers of Kneser graphs. https://doi.org/10.1002/jgt.22802

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