(H 1, H 2)-supermagic labelings for some shackles of connected graphs H 1 and H 2

Abstract

Let H ₁ and H ₂ be two graphs. A simple graph G = (V(G),E(G)) admits an (H ₁, H ₂)-covering, if every edge in E(G) belongs to at least one subgraph of G isomorphic to H ₁ or H ₂. The graph G is called (H ₁, H ₂)-magic, if there are two fixed positive integers k _i and k ₂, 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) = k ₁ and ∑ _v∈V(H″) f(v) + ∑ _e∈E(H″) f(e) = k ₂, for every subgraph H′ = (V(H′), E(H′)) of G isomorphic to H ₁ and for every subgraph H″ = (V(H), E(H)) of G isomorphic to H ₂. Moreover, it is said to be super (H ₁,H ₂)-magic, if f(V(G)) = {1, 2,…, |V(g)|}.

This paper aims to study an (H ₁, H ₂)-supermagic labelings for some shackles of connected graphs H ₁ and H ₂ such as cycle, flower, and prism graph. A shackle of G ₁, G ₂, G ₃,…, G_k denoted by shack(G ₁, G ₂, G ₃,…, G_k ) is a graph constructed from nontrivial connected and ordered graphs suchthat for every i and j ∈ [1, k] with |i − j| > 2, G _i and G_j have no common vertex, and for every i ∈ [1, k − 1], G_i and G _i+1 share exactly one common vertex, called linkage vertex, where the k − 1 linkage vertices are all distinct. In case G _i is isomorphic to H ₁ for odd i and G_i is isomorphic to H ₂ for even i, we denote such shackle by shack(H ₁, H ₂, k). We give a sufficient condition for some shack(H ₁, H ₂, k) being (H ₁, H ₂)-supermagic for even k.