Graph labelings in elementary abelian groups Original Research Article

Let p be an odd prime and let n ⩾ 1 be an integer. We show that if G is a directed graph without isolated vertices such that V(G)| ⩽ pn, then there exists an injective mapping ϕ from V(G) to the elementary abelian p-group of order pn such that ∑xϵV(C) ϕ(x) = 0 for every connected component C of G.