Abstract
In a proper edge-coloring of a cubic graph, an edge e is normal if the set of colors used by the five edges incident with an end of 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 μ3 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 |E(G)|−μ3(G) (which is no less than 2735|E(G)|) edges are normal. This result improves on some earlier results of Bílková and Šámal.
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