Further studies on chromatic completion of graphs

The chromatic completion graph of [Formula: see text] with respect to a proper vertex coloring [Formula: see text] of [Formula: see text], denoted by [Formula: see text], is the graph obtained by adding all possible edges to [Formula: see text] without violating the proper coloring protocol. The maximum number of edges added to [Formula: see text] to obtain the chromatic completion graph is the chromatic completion number [Formula: see text]. Equitable chromatic completion graph [Formula: see text] of a graph [Formula: see text] and equitable chromatic completion number [Formula: see text] are the equitable analogues of [Formula: see text] and [Formula: see text], respectively. In this paper, we present various structural aspects of chromatic completion graphs and equitable chromatic completion graphs. Also, the chromatic completion and the related parameter are described in terms of adjacency matrix and color matrix of graphs. The equitable chromatic completion graph is shown to be a Turán graph. More relevantly, we obtained the equitable chromatic completion number of an arbitrary graph [Formula: see text].