Graph coloring using commuting order product prime graph

Volume 23, Issue 2, pp 155--169 http://dx.doi.org/10.22436/jmcs.023.02.08

Publication Date: October 24, 2020 Submission Date: March 08, 2020 Revision Date: August 07, 2020 Accteptance Date: August 18, 2020

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Authors

Muhammed Bello - Department of Mathematics and Computer Science, Federal University of Kashere, P.M.B 0182 Gombe, Gombe State, Nigeria. Nor Muhainiah Mohd Ali - Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor, Bahru, Johor, Malaysia. Surajo Ibrahim Isah - Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor, Bahru, Johor, Malaysia.

Abstract

The concept of graph coloring has become a very active field of research that enhances many practical applications and theoretical challenges. Various methods have been applied in carrying out this study. Let \( G \) be a finite group. In this paper, we introduce a new graph of groups, which is a commuting order product prime graph of finite groups as a graph having the elements of \( G \) as its vertices and two vertices are adjacent if and only if they commute and the product of their order is a prime power. This is an extension of the study for order product prime graph of finite groups. The graph's general presentations on dihedral groups, generalized quaternion groups, quasi-dihedral groups, and cyclic groups have been obtained in this paper. Moreover, the commuting order product prime graph on these groups has been classified as connected, complete, regular, or planar. These results are used in studying various and recently introduced chromatic numbers of graphs.

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Keywords

• Commuting order product prime graph
• order product prime graph
• chromatic number
• graph coloring

MSC

•  05C25
•  20F65

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