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arXiv · 1611.02341

New bounds for locally irregular chromatic index of bipartite and subcubic graphs

Borut Lužar·Jakub Przybyło·Roman Soták
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Abstract

A graph is \textit{locally irregular} if the neighbors of every vertex $v$ have degrees distinct from the degree of $v$. \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. Recently, Bensmail et al. (Bensmail, Merker, Thomassen: Decomposing graphs into a constant number of locally irregular subgraphs, {\em European J. Combin.}, 60:124--134, 2017) settled the first constant upper bound for the problem to $328$ colors. In this paper, 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. In addition, we also prove that $4$ colors suffice for locally irregular edge-coloring of any subcubic graph.

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Borut Lužar, Jakub Przybyło, Roman Soták. 2016-11-07. New bounds for locally irregular chromatic index of bipartite and subcubic graphs. https://doi.org/10.1007/s10878-018-0313-7

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