Some cycle-supermagic labelings of the calendula graphs

Abstract

In this paper, we introduce a calendula graph, denoted by Clm,n. It is a graph constructed from a cycle on m vertices Cm and m copies of Cn which are Cn1, Cn2, ⋯, Cnm and grafting the i-th edge of Cm to an edge of in Cni for each i ∈ {1,2,⋯,m}. A graph G = (V, E) admits a Cn-covering, if every edge e ∈ E(G) belongs to a subgraph of G isomorphic to Cn. The graph G is called cycle-magic, if there exists a total labeling ϕ: V ∪ E → {1,2,…,|V|+|E|} such that for every subgraph Cn′ = (V′,E′) of G isomorphic to Cn has the same weight. In this case, the weight of Cn, denoted by ϕ(Cn’), is defined as ∑v∈V(C’n)ϕ(v) + ∑e∈E(C’n)ϕ(e). Furthermore, G is called cycle–supermagic, if ϕ:V→{1,2,…,|V|}. In this paper, we provide some cycle-supermagic labelings of calendula graphs. In order to prove it, we develop a technique, to make a partition of a multiset into m sub-multisets with the same cardinality such that the sum of all elements of each sub-multiset is same. The technique is called an m-balanced multiset.