A tight bound on the irregularity strength of graphs

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 oo if no such assignment is possible. Let G not equal K-3 be a graph on n vertices, with s(G) < infinity. Aigner and Triesch [SIAM J. Discrete Math, 3 (1990), pp. 439-449] used the congruence method to construct irregular assignments, showing s(G) less than or equal to n -1 if G is connected and s(G) less than or equal to n + 1 in general. We refine the congruence method in the disconnected case and show that s(G) less than or equal to n - 1 holds for all graphs with s(G) finite, except for K-3. This is tight and settles a conjecture of Aigner and Triesch.