On star coloring of modular product of graphs

Authors : Kaliraj K, Sivakami R, Vernold VIVIN J

Pages : 1235-1239

Doi:10.31801/cfsuasmas.768497

View : 9 | Download : 6

Publication Date : 2020-12-31

Article Type : Research Paper

Abstract :A star coloring of a graph $G$ is a proper vertex coloring in which every path on four vertices in $G$ is not bicolored. The star chromatic number $\chi_{s}\leftinsert ignore into journalissuearticles values(G\right);$ of $G$ is the least number of colors needed to star color $G$. In this paper, we find the exact values of the star chromatic number of modular product of complete graph with complete graph $K_m \diamond K_n$, path with complete graph $P_m \diamond K_n$ and star graph with complete graph $K_{1,m}\diamond K_n$. \par All graphs in this paper are finite, simple, connected and undirected graph and we follow \cite{bm, cla, f} for terminology and notation that are not defined here. We denote the vertex set and the edge set of $G$ by $Vinsert ignore into journalissuearticles values(G);$ and $Einsert ignore into journalissuearticles values(G);$, respectively. Branko Gr\`{u}nbaum introduced the concept of star chromatic number in 1973. A star coloring \cite{alberton, fertin, bg} of a graph $G$ is a proper vertex coloring in which every path on four vertices uses at least three distinct colors. The star chromatic number $\chi_{s}\leftinsert ignore into journalissuearticles values(G\right);$ of $G$ is the least number of colors needed to star color $G$. \par During the years star coloring of graphs has been studied extensively by several authors, for instance see \cite{alberton, col, fertin}.
Keywords : Star Coloring, Modular Product, Star graph