Normal edge‐colorings of cubic graphs

A normal k-edge-coloring of a cubic graph is a proper 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 to it, we have either exactly five distinct colors or exactly three distinct colors. We denote by chi N '(G) the smallest k, for which G admits a normal k-edge-coloring. Normal k-edge-colorings were introduced by Jaeger to study his well-known Petersen Coloring Conjecture. More precisely, it is known that proving chi N '(G)<= 5 for every bridgeless cubic graph is equivalent to proving the Petersen Coloring Conjecture and then it implies, 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 chi N '(G)=7. In contrast, the known best general upper bound for chi N '(G) was 9. Here, we improve it by proving that chi N '(G)<= 7 for any simple cubic graph G, which is best possible. We obtain this result by proving the existence of specific nowhere zero Z22-flows in 4-edge-connected graphs.