Abstract
A graph G admits an H-covering if every edge of G belongs to a subgraph isomorphic to a given graph H. G is said to be H-magic if there exists a bijection f:V(G)∪E(G)→{1,2,…,|V(G)|+|E(G)|} such that wf(H′)=∑v∈V(H′)f(v)+∑e∈E(H′)f(e) is a constant, for every subgraph H′ isomorphic to H. In particular, G is said to be H-supermagic if f(V(G))={1,2,…,|V(G)|}. When H is isomorphic to a complete graph K2, an H-(super)magic labeling is an edge-(super)magic labeling. Suppose that G admits an F-covering and H-covering for two given graphs F and H. We define G to be (F,H)-sim-(super)magic if there exists a bijection f′ that is simultaneously F-(super)magic and H-(super)magic. In this paper, we consider (K2,H)-sim-(super)magic where H is isomorphic to three classes of graphs with varied symmetry: a cycle which is symmetric (both vertex-transitive and edge-transitive), a star which is edge-transitive but not vertex-transitive, and a path which is neither vertex-transitive nor edge-transitive. We discover forbidden subgraphs for the existence of (K2,H)-sim-(super)magic graphs and classify classes of (K2,H)-sim-(super)magic graphs. We also derive sufficient conditions for edge-(super)magic graphs to be (K2,H)-sim-(super)magic and utilize such conditions to characterize some (K2,H)-sim-(super)magic graphs.
Highlights
G is said to be edge-magic (EMT) and k is called a magic sum. This notion was defined by Kotzig and Rosa [1], who called it magic valuation, and later rediscovered by Ringel and Lladó [2]
In 1998, Enomoto et al [3] introduced a special case of EMT labeling with the extra property that λ(V ( G )) = {1, 2, . . . , |V ( G )|}
An SEMT labeling has a significant role in graph labeling because it is related to other types of labelings
Summary
The theorem stated forbidden subgraphs of (K2 , Pn )-sim-(super)magic graphs. Assume to the contrary that G is (K2 , Pn )-sim-(super)magic and contains a subgraph isomorphic to Amal ((m, 3)-(tadpole), Pn−m ; H; 2). A graph G is (K2 , P3 )-sim-(super)magic if and only if G is isomorphic to the star Sn for any positive integer n ≥ 3.
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