Sharp Bounds and Precise Values for the Ni-Chromatic Number of Graphs

Abstract

Let [Formula: see text] be a connected undirected graph. A vertex coloring [Formula: see text] of [Formula: see text] is an [Formula: see text]-vertex coloring if for each vertex [Formula: see text] in [Formula: see text], the number of different colors assigned to [Formula: see text] is at most [Formula: see text]. The [Formula: see text]-chromatic number of [Formula: see text], denoted by [Formula: see text], is the maximum number of colors which are used in an [Formula: see text]-vertex coloring of [Formula: see text]. In this paper, we provide sharp bounds for [Formula: see text] of a graph [Formula: see text] in terms of its vertex cover number, maximum degree and diameter, respectively. We also determine precise values for [Formula: see text] in some cases.