Abstract
The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-coloring in which every color class contains at least one vertex adjacent to some vertex in all the other color classes. It is proved that with four exceptions, the b-chromatic number of cubic graphs is 4. The exceptions are the Petersen graph, K 3,3, the prism over K 3, and one more sporadic example on 10 vertices.
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Jakovac, M., Klavžar, S. The b-Chromatic Number of Cubic Graphs. Graphs and Combinatorics 26, 107–118 (2010). https://doi.org/10.1007/s00373-010-0898-9
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DOI: https://doi.org/10.1007/s00373-010-0898-9