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

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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.

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Acknowledgements

Borut Lužar was partly supported by the Slovenian Research Agency Program P1–0383 and by the National Scholarship Programme of the Slovak Republic. Jakub Przybyło was supported by the National Science Centre, Poland, Grant No. 2014/13/B/ST1/01855 and partly supported by the Faculty of Applied Mathematics AGH UST statutory tasks within subsidy of Ministry of Science and Higher Education. Roman Soták was supported by the Slovak Research and Development Agency under the Contract No. APVV–15–0116 and by the Slovak VEGA Grant 1/0368/16.

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Authors and Affiliations

  1. Faculty of Information Studies, Novo Mesto, Ljubljanska cesta 31A, 8000, Novo Mesto, Slovenia

    Borut Lužar

  2. Faculty of Science, Pavol Jozef Šafárik University, Jesenná 566/5, 040 01, Kosice, Slovakia

    Borut Lužar & Roman Soták

  3. AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059, Kraków, Poland

    Jakub Przybyło

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  1. Borut Lužar
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  2. Jakub Przybyło
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  3. Roman Soták
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Correspondence to Borut Lužar.

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Lužar, B., Przybyło, J. & Soták, R. New bounds for locally irregular chromatic index of bipartite and subcubic graphs. J Comb Optim 36, 1425–1438 (2018). https://doi.org/10.1007/s10878-018-0313-7

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  • DOI: https://doi.org/10.1007/s10878-018-0313-7

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