Integral sum graphs Gn and G-r,n are perfect graphs

Abstract

A graph G is an integral sum graph (sum graph) if its vertices can be labeled with distinct integers (positive integers) so that e = uv is an edge of G if and only if the sum of the labels on vertices u and v is also a label in G. A graph G is perfect if the chromatic number of each of its induced subgraphs is equal to the clique number of the same. A simple graph G is of class 1 if its edge chromatic number and maximum degree are same. In this paper, we prove that integral sum graphs Gn , G 0 , n and G − r , n over the label sets [ 1 , n ] , [ 0 , n ] and [ − r , n ] , respectively, are perfect graphs as well as of class 1 for r , n ∈ N . We also obtain a few structural properties of these graphs.