Vertex neighborhood restricted edge achromatic sums of graphs

Abstract

The vertex induced 2-edge coloring number [Formula: see text] of a graph [Formula: see text] is the highest number of colors that can occur in an edge coloring of a graph [Formula: see text] such that not more than two colors can be used to color the edges in the induced subgraph [Formula: see text] generated by the closed neighborhood [Formula: see text] of a vertex [Formula: see text] in [Formula: see text]. The vertex induced 2-edge coloring sum of a graph [Formula: see text] denoted as [Formula: see text], is the greatest sum among all the vertex induced 2-edge coloring of a graph [Formula: see text] which concedes [Formula: see text] colors. The vertex incident 2-edge coloring number of a graph [Formula: see text] is the highest number of colors required to color the edges of a graph [Formula: see text] such that not more than two colors can be ceded to the edges incident at the vertex [Formula: see text] of [Formula: see text]. The vertex incident 2-edge coloring sum of a graph [Formula: see text] denoted as [Formula: see text], is the maximum sum among all the vertex incident 2-edge coloring of graph [Formula: see text] which receives maximum [Formula: see text] colors. In this paper, we initiate a study on the vertex induced 2-edge coloring sum and vertex incident 2-edge coloring sum concepts and apply the same to some graph classes. Besides finding the exact values of these parameters, we also obtain some bounds and a few comparative results.