3-Sequent achromatic sum of graphs

Abstract

Three vertices [Formula: see text] in a graph [Formula: see text] are said to be [Formula: see text]-sequent if [Formula: see text] and [Formula: see text] are adjacent edges in [Formula: see text]. A 3-sequent coloring (3s coloring) is a function [Formula: see text] such that if [Formula: see text] and [Formula: see text] are 3-sequent vertices, then either [Formula: see text] or [Formula: see text] (or both). The [Formula: see text]-sequent achromatic number of a graph [Formula: see text], denoted [Formula: see text], equals the maximum number of colors that can be used in a coloring of the vertices’ of [Formula: see text] such that if [Formula: see text] and [Formula: see text] are any two sequent edges in [Formula: see text], then either [Formula: see text] or [Formula: see text] is colored the same as [Formula: see text]. The [Formula: see text]-sequent achromatic sum of a graph [Formula: see text], denoted [Formula: see text], is the greatest sum of colors among all proper 3s-coloring that requires [Formula: see text] colors. This research initiates the study of [Formula: see text]-sequent achromatic sum and finds the exact values of this parameter for some known graphs. Furthermore, we calculate the [Formula: see text] of corona product, Cartesian product of the graphs and some important results have been proved and a comparative study is carried out.