Partially normal 5-edge-colorings of cubic graphs

In a proper edge-coloring of a cubic graph, an edge $e$ is normal if the set of colors used by the edges adjacent to $e$ has cardinality 3 or 5. The Petersen coloring conjecture asserts that every bridgeless cubic graph has a normal 5-edge-coloring, that is, a proper 5-edge-coloring such that all edges are normal. In this paper, we prove a result related to the Petersen coloring conjecture. The parameter ${\mu }_3$ is a measurement for cubic graphs, introduced by Steffen in 2015. Our result shows that every bridgeless cubic graph $G$ has a proper 5-edge-coloring such that at least $\mathrm{\mid }E(G)\mathrm{\mid }-{\mu }_3(G)$, which is no less than $\frac{4}{5}\mathrm{\mid }E(G)\mathrm{\mid }$, many edges are normal. This result improves on some earlier results of B'{\i}lkov'{a} and \v{S}'{a}mal.