A Tight Bound on the Irregularity Strength of Graphs

Abstract

An assignment of positive integer weights to the edges of a simple graph G is called irregular if the weighted degrees of the vertices are different. The {irregularity strength} s(G) is the maximal weight, minimized over all irregular assignments. It is set to $\infty$ if no such assignment is possible. Let $G \neq K_3$ be a graph on n vertices, with s(G) < \infty$. Aigner and Triesch [SIAM J. Discrete Math, 3 (1990), pp. 439--449] used the congruence method to construct irregular assignments, showing $s(G) \le n-1$ if G is connected and $s(G) \le n+1$ in general. We refine the congruence method in the disconnected case and show that $s(G) \leq n-1$ holds for all graphs with s(G) finite, except for K3. This is tight and settles a conjecture of Aigner and Triesch.