Let Γ be an additive abelian group and G = (V, A) a directed graph, both of order n. A Γ labeling of G is a bijection ℓ: V → Γ. Given such a labeling ℓ, for each x in V, define w(x) to be the sum of the labels on the vertices of tails of arcs with head x minus the sum of the labels on the vertices that are heads of arcs with tail x. If ℓ is a constantfunction, then ℓ is said to be a directed Γ-distance magic labeling for G. A graph G is said to be orientable Γ-distance magic if there exists a directed graph G with underlying graph G and a directed Γ-distance magic labeling for G. It has been conjectured that every 2r-regular graph G of order n is orientable ℤn-distance magic. In this paper we find orientable ℤn-distance magic labelings of some products of graphs, namely the strong and lexicographic products. We provide orientable ℤn-distance magic labelings for some classes of regular and non-regular graphs which arise via these products, and we identify some graphs which are not orientable ℤn-distance magic.
|Original language||English (US)|
|Number of pages||10|
|Journal||Australasian Journal of Combinatorics|
|State||Published - Jan 1 2018|