Geodesic
On Total H-Irregularity Strength of the Disjoint Union of Graphs
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 181-194.

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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.
Keywords: H -covering, H -irregular labeling, total H -irregularity strength, copies of graphs, union of graphs 
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Ashraf, Faraha; López, Susana Clara; Muntaner-Batle, Francesc Antoni; Oshima, Akito; Bača, Martin; Semaničová-Feňovčíková, Andrea. On Total H-Irregularity Strength of the Disjoint Union of Graphs. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 181-194. http://geodesic.mathdoc.fr/item/DMGT_2020_40_1_a11/

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