New bounds for locally irregular chromatic index of bipartite and subcubic graphs
Lužar, Borut ; Przybyło, Jakub ; Soták, Roman
ǂA ǂgraph is ▫$\textit{locally irregular}$▫ if the neighbors of every vertex ▫$v$▫ have degrees distinct from the degree of ▫$v$▫. A ▫$\textit{locally irregular edge-coloring}$▫ of a graph ▫$G$▫ is ...an (improper) edge-coloring such that the graph induced on the edges of any color class is locally irregular. It is conjectured that ▫$3$▫ colors suffice for a locally irregular edge-coloring. In the paper, we develop a method using which we prove ▫$4$▫ colors are enough for a locally irregular edge-coloring of any subcubic graph admiting such a coloring. We believe that our method can be further extended to prove the tight bound of ▫$3$▫ colors for such graphs. Furthermore, using a combination of existing results, we present an improvement of the bounds for bipartite graphs and general graphs, setting the best upper bounds to 7 and 220, respectively.
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