Strong edge colorings of graphs and the covers of Kneser graphs

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 graphK(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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