(H1, H2)-supermagic labelings for some shackles of connected graphs H1 and H2

Abstract

Let H1 and H2 be two graphs. A simple graph G = (V(G),E(G)) admits an (H1, H2)-covering, if every edge in E(G) belongs to at least one subgraph of G isomorphic to H1 or H2. The graph G is called (H1, H2)-magic, if there are two fixed positive integers ki and k2, and a bijective function f : V(G) ∪ E(G) → {1, 2,…, |V(G)| + |E(G)|} suchthat ∑v∈V(H′) f(v) + ∑e∈E(H′) f(e) = k1 and ∑v∈V(H″) f(v) + ∑e∈E(H″) f(e) = k2, for every subgraph H′ = (V(H′), E(H′)) of G isomorphic to H1 and for every subgraph H″ = (V(H), E(H)) of G isomorphic to H2. Moreover, it is said to be super (H1,H2)-magic, if f(V(G)) = {1, 2,…, |V(g)|}.This paper aims to study an (H1, H2)-supermagic labelings for some shackles of connected graphs H1 and H2 such as cycle, flower, and prism graph. A shackle of G1, G2, G3,…, Gk denoted by shack(G1, G2, G3,…, Gk) is a graph constructed from nontrivial connected and ordered graphs suchthat for every i and j ∈ [1, k] with |i − j| > 2, Gi and Gj have no common vertex, and for every i ∈ [1, k − 1], Gi and Gi+1 share exactly one common vertex, called linkage vertex, where the k − 1 linkage vertices are all distinct. In case Gi is isomorphic to H1 for odd i and Gi is isomorphic to H2 for even i, we denote such shackle by shack(H1, H2, k). We give a sufficient condition for some shack(H1, H2, k) being (H1, H2)-supermagic for even k.