Some bounds on local connective Chromatic number

Abstract

Graph coloring is one of the most important concept in graph theory. Many practical problems can be formulated as graph coloring problems. In this paper, we define a new coloring concept called local connective coloring. A local connective k-coloring of a graph G is a proper vertex coloring, which assigns colors from {1, 2, ..., k} to the vertices V (G) in a such way that any two nonadjacent vertices u and v of a color i satisfies κ(u, v) i, where κ(u, v) is the maximum number of internally disjoint paths between u and v. Adjacent vertices are colored with different colors as in the proper coloring. The smallest integer k for which there exists a local connective k-coloring of G is called the local connective chromatic number of G, and it is denoted by χ lc (G). We study this coloring on several classes of graphs and give some general bounds. We also compare local connective chromatic number of a graph with chromatic number and packing chromatic number of it.