On Edge H-Irregularity Strengths of Some Graphs

Abstract

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: 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.