Indian Journal of Science and Technology

Abstract

Objectives: For a given graph with proper coloring, the problem of selecting a dom-coloring set is to choose a dominating set having a property that it has a minimum of one vertex from every possible color class in . Our aim is to determine the family of networks that allow dom-coloring and to find its dom-chromatic number denoted by . Method: We have applied the algorithmic method of choosing the dom-coloring set(dc-set). Here we have designed a coloring algorithm to yield the proper coloring for the vertices of the graph. D-set algorithm has been developed to determine the dominating set for the given graph. Then, the dc-set for the graph is obtained by applying the above two algorithms. Findings: In this study, we have established the study on finding the dc-set of wrapped butterfly network and bloom graphs. Further, we have found the dom-chromatic number of the above-mentioned graphs. Novelty: Dom-coloring is an extended variation of graph coloring and domination which has emerged as a result of the combination of the two broad concepts in graph theory namely, domination and coloring. A dominating set which includes a minimum of one vertex from all possible color classes of the graph forms a dom-coloring set. In this paper, a study on dom-coloring of wrapped butterfly and bloom graphs have been accomplished. These results may be generalized for butterfly derived networks to determine its dom-chromatic number.

Keywords: Dominating set, Domination, number, Coloring, Chromatic number, Dom­coloring set, Dom­chromatic number 1