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

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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References

  1. Addario-Berry L, Aldred REL, Dalal K, Reed BA (2005) Vertex colouring edge partitions. J Combin Theory Ser B 94(2):237–244

    MathSciNet  Article  Google Scholar 

  2. Baudon O, Bensmail J, Przybyło J, Woźniak M (2015) On decomposing regular graphs into locally irregular subgraphs. Eur J Combin 49:90–104

    MathSciNet  Article  Google Scholar 

  3. Baudon O, Bensmail J, Sopena É (2015) On the complexity of determining the irregular chromatic index of a graph. J Discret Algorithms 30:113–127

    MathSciNet  Article  Google Scholar 

  4. Bensmail J, Merker M, Thomassen C (2017) Decomposing graphs into a constant number of locally irregular subgraphs. Eur J Combin 60:124–134

    MathSciNet  Article  Google Scholar 

  5. Havet F, Paramaguru N, Sampathkumar R (2014) Detection number of bipartite graphs and cubic graphs. Discret Math Theor Comput Sci 16(3):333–342

    MathSciNet  MATH  Google Scholar 

  6. Kalkowski M, Karoński M, Pfender F (2010) Vertex-coloring edge-weightings: towards the 1-2-3-conjecture. J Combin Theory Ser B 100(3):347–349

    MathSciNet  Article  Google Scholar 

  7. Karoński M, Łuczak T, Thomason A (2004) Edge weights and vertex colours. J Combin Theor Ser B 91(1):151–157

    MathSciNet  Article  Google Scholar 

  8. Lužar B, Petruševski M, Škrekovski R (2018) On vertex-parity edge-colorings. J Combin Optim 35:373–388

    MathSciNet  Article  Google Scholar 

  9. Przybyło J (2016) On decomposing graphs of large minimum degree into locally irregular subgraphs. Electron J Combin 23:2–31

    MathSciNet  MATH  Google Scholar 

Download references

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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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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Keywords

  • Locally irregular graph
  • Locally irregular edge-coloring
  • Bipartite graph
  • Subcubic graph