Enumerating super edge-magic labelings for the union of non-isomorphic graphs

A super edge-magic labeling of a graph G=(V,E) of order p and size q is a bijection f:V ∪E→{i}p+qi=1 such that: (1) f(u)+f(uv)+f(v)=k for all uv∈E; and (2) f(V )={i}pi=1. Furthermore, when G is a linear forest, the super edge-magic labeling of G is called strong if it has the extra property that if uv∈E(G) , u′,v′ ∈V (G) and dG (u,u′ )=dG (v,v′ )<+∞, then f(u)+f(v)=f(u′ )+f(v′ ). In this paper we introduce the concept of strong super edge-magic labeling of a graph G with respect to a linear forest F, and we study the super edge-magicness of an odd union of nonnecessarily isomorphic acyclic graphs. Furthermore, we find exponential lower bounds for the number of super edge-magic labelings of these unions. The case when G is not acyclic will be also considered.Preprin

On super edge-magic decomposable graphs

Let G be any graph and let {Hi}i∈I be a family of graphs such that E(Hi) ∩ E(Hj ) = ∅ when i 6= j, ∪i∈IE(Hi) = E(G) and E(Hi) 6= ∅ for all i ∈ I. In this paper we introduce the concept of {Hi}i∈I -super edge-magic decomposable graphs and {Hi}i∈I -super edge-magic labelings. We say that G is {Hi}i∈I -super edge-magic decomposable if there is a bijection β : V (G) → {1, 2, . . . , |V (G)|} such that for each i ∈ I the subgraph Hi meets the following two requirements: β(V (Hi)) = {1, 2, . . . , |V (Hi)|} and {β(a) + β(b) : ab ∈ E(Hi)} is a set of consecutive integers. Such function β is called an {Hi}i∈I -super edge-magic labeling of G. We characterize the set of cycles Cn which are {H1, H2}-super edge-magic decomposable when both, H1 and H2 are isomorphic to (n/2)K2. New lines of research are also suggested