A normal k-edge-coloring of a cubic graph is an edge-coloring with k
colors having the additional property that when looking at the set of colors
assigned to any edge e and the four edges adjacent it, we have either exactly
five distinct colors or exactly three distinct colors. We denote by
χN′(G) the smallest k, for which G admits a normal
k-edge-coloring. Normal k-edge-colorings were introduced by Jaeger in order
to study his well-known Petersen Coloring Conjecture. More precisely, it is
known that proving χN′(G)≤5 for every bridgeless cubic graph is
equivalent to proving Petersen Coloring Conjecture and then, among others,
Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture. Considering the
larger class of all simple cubic graphs (not necessarily bridgeless), some
interesting questions naturally arise. For instance, there exist simple cubic
graphs, not bridgeless, with χN′(G)=7. On the other hand, the known
best general upper bound for χN′(G) was 9. Here, we improve it by
proving that χN′(G)≤7 for any simple cubic graph G, which is best
possible. We obtain this result by proving the existence of specific no-where
zero Z22-flows in 4-edge-connected graphs.Comment: 17 pages, 6 figure