Irregularity strength and compound graphs (Q1978158)

The irregularity strength of a graph is the smallest integer \(k\) for which the edges may be assigned weights not exceeding \(k\), and which sum at each vertex into distinct numbers. A compound graph of two graphs \(H\) and \(G\) is obtained by replacing each vertex of \(H\) by a copy of \(G\) and adding some perfect matching between any two copies which were linked in \(H\). An upper bound is derived for the irregularity strength of a compound graph, leading to determination of the exact irregularity strength for two particular families of graphs.