An application of the combinatorial Nullstellensatz to a graph labelling problem

An antimagic labelling of a graph G with m edges and n vertices, is a bijection from the set of edges of G to the set of integers {1, . . . ,m}, such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with that vertex. A graph is called antimagic if it admits an antimagic labelling. In [7], Ringel has conjectured that every simple connected graph, other than K2, is antimagic. In this paper, we prove a special case of this conjecture. Namely, we prove that if G is a graph on n = p vertices, where p is an odd prime and k is a positive integer, that admits a Cp-factor, then it is antimagic. The case p = 3 was proved in [8]. Our main tool is the Combinatorial Nullstellensatz (c.f. [1]).