F. Ashraf, M. Bača, S.C. López , F.A. Muntaner-Batle, A. Oshima, A. Semaničová-Feňovčiková
On total H-irregularity strength of the disjoint union of graphs
Discussiones Mathematicae Graph Theory
Received 12.04.2017, Revised 05.02.2018, Accepted 22.02.2018, doi: 10.7151/dmgt.2118

A simple graph G admits an H-covering if every edge in E(G) belongs to at least to one subgraph of G isomorphic to a given graph H. For the subgraph H⊆ G under a total k-labeling we define the associated H-weight as the sum of labels of all vertices and edges belonging to H. The total k-labeling is called the H-irregular total k-labeling of a graph G admitting an H-covering if all subgraphs of G isomorphic to H have distinct weights. The total H-irregularity strength of a graph G is the smallest integer k such that G has an H-irregular total k-labeling. In this paper, we estimate lower and upper bounds on the total H-irregularity strength for the disjoint union of multiple copies of a graph and the disjoint union of two non-isomorphic graphs. We also prove the sharpness of the upper bounds.
H-covering, H-irregular labeling, total H-irregularity strength, copies of graphs, union of graphs