On the b-chromatic number of regular graphs

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 that has a neighbor in each of the other color classes. We prove that every d-regular graph with at least 2d 3 vertices has b-chromatic number d + 1, that the b-chromatic number of an arbitrary d-regular graph with girth g = 5 is at least d+1 2 and that every d-regular graph, d ≥ 6, with diameter at least d and with no 4-cycles admits a b-coloring with d + 1 colors.

Coloring sums of extensions of certain graphs

“…This result has been improved in [3] by bounding the result to graphs having at least 2r 3 vertices. So it is easy to see that for C n , n ≥ 54, we have, ϕ(C n ) = 3 and ϕ(C x n,c ) = 4.…”

The b-chromatic index of direct product of graphs

“…If G is an r-edge regular graph, then L(G) is an r-regular graph. ''Usually'' this means that ϕ ′ (G) = r + 1; in particular, it holds for all r-edge regular graphs with at least 2r 3 edges (Theorem 2.2 in [5]). But even if the number of edges is smaller than 2r 3 , one can expect that most problems will occur when the number of edges is small.…”

“…While the reader might find that this corollary holds only for a small number of graphs, let us recall that there exist only a finite number of r-regular graphs with ϕ(G) < m(G) (see [5]). If G is an r-edge regular graph, then L(G) is an r-regular graph.…”

“…This bound was later improved to 2d 3 in [5]. In particular, it was shown in [18] that there are only four exceptions among cubic graphs with ϕ(G) < 4, one of them being the Petersen graph.…”

A characterization of b-chromatic and partial Grundy numbers by induced subgraphs

“…[8,19]). The b-chromatic number of regular graphs has been investigated in a series of papers [6,17,20,22]. Determining the b-chromatic number of a tight graph is NP-hard even for a connected bipartite graph [18] and a tight chordal graph [12].…”