Abstract
A b-coloring of a graph is a proper coloring where each color admits at least one node (called dominating node) adjacent to every other used color. The maximum number of colors needed to b-color a graph G is called the b-chromatic number and is denoted by φ(G). In this paper, we find the b-chromatic number and some of the structural properties of corona product of crown graph and complete bipartite graphwith path graph.
Highlights
A b-coloring by k-colors is a proper coloring of the vertices of graph G such that in each color classes there exists a vertex that has neighbors in all the other k-1 color classes
We find the b-chromatic number and some of the structural properties of corona product of crown graph and complete bipartite graph with path graph
The corona G1 ◦ G2 of two graphs G1 and G2 is defined as a graph obtained by taking one copy of G1 and p1 copies of G2 and attach one copy of G2 at every vertex of G1 (Harary, 1972)
Summary
A b-coloring by k-colors is a proper coloring of the vertices of graph G such that in each color classes there exists a vertex that has neighbors in all the other k-1 color classes. The b-chromatic number φ(G) is the largest number k for which G admits a bcoloring with k-colors (Irving and Manlove, 1999). The corona G1 ◦ G2 of two graphs G1 and G2 is defined as a graph obtained by taking one copy of G1 (which has p1 vertices) and p1 copies of G2 and attach one copy of G2 at every vertex of G1 (Harary, 1972). In this paper we find for which the largest number k for which corona product of crown graph and complete bipartite graph with path graph admits a b-coloring with k-colors. We find some of its structural properties (Venkatachalam and Vernold Vivin, 2010; Vernold Vivin and Venkatachalam, 2012; Vijayalakshmi and Thilagavathi, 2012)
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