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

Borut Lužar (), Jakub Przybyło () and Roman Soták ()
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Borut Lužar: Faculty of Information Studies, Novo Mesto
Jakub Przybyło: AGH University of Science and Technology
Roman Soták: Pavol Jozef Šafárik University

Journal of Combinatorial Optimization, 2018, vol. 36, issue 4, No 17, 1425-1438

Abstract: Abstract A graph is locally irregular if the neighbors of every vertex v have degrees distinct from the degree of v. A 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 three colors suffice for a locally irregular edge-coloring. In the paper, we develop a method using which we prove four 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 three 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.

Keywords: Locally irregular graph; Locally irregular edge-coloring; Bipartite graph; Subcubic graph (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10878-018-0313-7

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