Normal edge‐colorings of cubic graphs

Abstract

AbstractA normal ‐edge‐coloring of a cubic graph is a proper edge‐coloring with colors having the additional property that when looking at the set of colors assigned to any edge and the four edges adjacent to it, we have either exactly five distinct colors or exactly three distinct colors. We denote by the smallest , for which admits a normal ‐edge‐coloring. Normal ‐edge‐colorings were introduced by Jaeger to study his well‐known Petersen Coloring Conjecture. More precisely, it is known that proving 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 . In contrast, the known best general upper bound for was . Here, we improve it by proving that for any simple cubic graph , which is best possible. We obtain this result by proving the existence of specific nowhere zero ‐flows in ‐edge‐connected graphs.