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=K3,3); we conjecture that the irregularity strength of an r-regular graph of order n⩽2r is 3, except if G is Kl,l with l odd.