Locally irregular edge-coloring of subcubic graphs

10/10/2022

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by   Borut Lužar, et al.

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A graph is locally irregular if no two adjacent vertices have the same degree. A locally irregular edge-coloring of a graph G is such an (improper) edge-coloring that the edges of any fixed color induce a locally irregular graph. Among the graphs admitting a locally irregular edge-coloring, i.e., decomposable graphs, only one is known to require 4 colors, while for all the others it is believed that 3 colors suffice. In this paper, we prove that decomposable claw-free graphs with maximum degree 3, all cycle permutation graphs, and all generalized Petersen graphs admit a locally irregular edge-coloring with at most 3 colors. We also discuss when 2 colors suffice for a locally irregular edge-coloring of cubic graphs and present an infinite family of cubic graphs of girth 4 which require 3 colors.