  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.
