Irregularity strength of regular graphs of large degree

Abstract

As defined by Chartrand et al. [2], the irregularity strength of a graph G is the smallest possible value of k for which we can assign positive integers not greater than k to the edges of G in such a way that the sums at each vertex are distinct. We prove that, if G is an ( n−3)- or ( n−4)- regular graph of order n, the strength of G is 3 (except if G= K 3,3); we conjecture that the irregularity strength of an r-regular graph of order n⩽2 r is 3, except if G is K l,l with l odd.