We investigate the modular edge-gracefulness k(G) of a graph, i.e., the least integer k such that taking a cyclic group Zk of order k, there exists a function f:E(G)→Zk so that the sums of edge labels incident with every vertex are distinct. So far the best upper bound on k(G) for a general graph G is 2n, where n is the order of G. In this note we prove that if G is a graph of order n without star as a component then k(G)=n for n¬≡2(mod4) and k(G)=n+1 otherwise. Moreover we show that for such G for every integer t≥k(G) there exists a Zt-irregular labeling.

Hefetz, Mütze, and Schwartz conjectured that every connected undirected graph admits an antimagic orientation (Hefetz et al., 2010). In this paper we support the analogous question for distance magic labeling. Let Γ be an Abelian group of order n. A directed Γ-distance magic labeling of an oriented graph ⃗ G = (V, A) of order n is a bijection ⃗ l : V → Γ with the property that there is a magic constant µ ∈ Γ such that for every x ∈ V (G) w(x) = ∑ y∈N + (x) ⃗ l(y) − ∑ y∈N − (x) ⃗ l(y) = µ. In this paper we provide an infinite family of odd regular graphs possessing an orientable Z n-distance magic labeling. Our results refer to lexicographic product of graphs. We also present a family of odd regular graphs that are not orientable Z n-distance magic.

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