Dense Graphs are Antimagic

An of a graph with edges and vertices is a bijection from the set of edges to the integers 1 such that all vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called if it has an antimagic labeling. A conjecture of Ringel (see [4]) states that every connected graph, but , is antimagic. Our main result validates this conjecture for graphs having minimum degree (log ). The proof combines probabilistic arguments with simple tools from analytic number theory and combinatorial techniques. We also prove that complete partite graphs (but ) and graphs with maximum degree at least 2 are antimagic