On the irregular coloring of bipartite graph and tree graph families

Abstract

Abstract This article discusses irregular coloring. Irregular coloring was first introduced by Mary Radcliffe and Ping Zhang in 2007. The coloring c is called irregular coloring if distinct vertices of G have distinct codes. The color code of a vertex v of G with respect to c is code(v) = (a0, a1, a2,…, a k) = a0a1a2, …ak, where a0 = c(v) and ai, (1 < i < k) is the number of vertices that are adjacent to v and colored i. The minimum k-color used in irregular coloring is called the irregular chromatic number and is denoted by \ir. Irregular coloring is included in proper coloring, where each vertex that is the neighbors must not be the same color. The graphs used in this article are a family of bipartite graphs and a family of tree graphs, including complete bipartite graphs, crown graphs, star graphs, centipede graphs, and double star graphs.