The irregularity strength of tKp

Assign positive integer weights to the edges of a simple graph G (with no isolated edges and vertices) of order at least 3 in such a way that the graph becomes irregular, i.e. the weight sums at the vertices become pairwise distinct. The minimum of the largest weights assigned over all such irregular assignments on the union of t copies of the complete graph with p vertices, p ⩾ 3, is determined. A recent conjecture of Faudree et al. is disproved.