Geodesic
On Edge H-Irregularity Strengths of Some Graphs
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 4, pp. 949-961

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For a graph G an edge-covering of G is a family of subgraphs H1, H2, . . ., Ht such that each edge of E(G) belongs to at least one of the subgraphs Hi, i = 1, 2, . . ., t. In this case we say that G admits an (H1, H2, . . ., Ht)-(edge) covering. An H-covering of graph G is an (H1, H2, . . ., Ht)-(edge) covering in which every subgraph Hi is isomorphic to a given graph H. Let G be a graph admitting H-covering. An edge k-labeling α : E(G) → 1, 2, . . ., k is called an H-irregular edge k-labeling of the graph G if for every two different subgraphs H′ and H′′ isomorphic to H their weights wtα(H′) and wtα(H′″) are distinct. The weight of a subgraph H under an edge k-labeling is the sum of labels of edges belonging to H. The edge H-irregularity strength of a graph G, denoted by ehs(G, H), is the smallest integer k such that G has an H-irregular edge k-labeling. In this paper we determine the exact values of ehs(G, H) for prisms, antiprisms, triangular ladders, diagonal ladders, wheels and gear graphs. Moreover the subgraph H is isomorphic to only C4, C3 and K4.
Keywords: prism, antiprism, triangular ladder, diagonal ladder, wheel, gear graph, H-irregular edge labeling, edge H-irregularity strength 
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     author = {Naeem, Muhammad and Siddiqui, Muhammad Kamran and Ba\v{c}a, Martin and Semani\v{c}ov\'a-Fe\v{n}ov\v{c}{\'\i}kov\'a, Andrea and Ashraf, Faraha},
     title = {On {Edge} {H-Irregularity} {Strengths} of {Some} {Graphs}},
     journal = {Discussiones Mathematicae. Graph Theory},
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Naeem, Muhammad; Siddiqui, Muhammad Kamran; Bača, Martin; Semaničová-Feňovčíková, Andrea; Ashraf, Faraha. On Edge H-Irregularity Strengths of Some Graphs. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 4, pp. 949-961. http://geodesic.mathdoc.fr/item/DMGT_2021_41_4_a5/