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