The irregularity strength and cost of the union of cliques

Abstract

Assign positive integer weights to the edges of a simple graph with no component isomorphic to Ki or K2, 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 vertex-disjoint union of complete graphs is determined. The method of proof also yields the smallest possible total increase in the sum of edge weights in irregular asignments, called irregularity cost.