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The b-Chromatic Index of a Graph

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Abstract

The b-chromatic index \(\varphi '(G)\) of a graph \(G\) is the largest integer \(k\) such that \(G\) admits a proper \(k\)-edge coloring in which every color class contains at least one edge incident to some edge in all the other color classes. The b-chromatic index of trees is determined and equals either to a natural upper bound \(m'(T)\) or one less, where \(m'(T)\) is connected with the number of edges of high degree. Some conditions are given for which graphs have the b-chromatic index strictly less than \(m'(G)\), and for which conditions it is exactly \(m'(G)\). In the last part of the paper, regular graphs are considered. It is proved that with four exceptions, the b-chromatic index of cubic graphs is \(5\). The exceptions are \(K_4\), \(K_{3,3}\), the prism over \(K_3\), and the cube \(Q_3\).

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Acknowledgments

This work has been financed by ARRS Slovenia under the Grant P1-0297 and within the EUROCORES Programme EUROGIGA (Project GReGAS) of the European Science Foundation.

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Authors and Affiliations

  1. Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000, Maribor, Slovenia

    Marko Jakovac

  2. Faculty of Electrical Engineering and Computer Science, University of Maribor, Smetanova ulica 17, 2000, Maribor, Slovenia

    Iztok Peterin

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  1. Marko Jakovac
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Correspondence to Marko Jakovac.

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Communicated by Xueliang Li.

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Jakovac, M., Peterin, I. The b-Chromatic Index of a Graph. Bull. Malays. Math. Sci. Soc. 38, 1375–1392 (2015). https://doi.org/10.1007/s40840-014-0088-7

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  • DOI: https://doi.org/10.1007/s40840-014-0088-7

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