 
Summary: Dense graphs are antimagic
N. Alon
G. Kaplan
A. Lev
Y. Roditty §
R. Yuster ¶
Abstract
An antimagic labeling of a graph with m edges and n vertices is a bijection from the set of
edges to the 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 the same vertex. A graph is called antimagic if
it has an antimagic labeling. A conjecture of Ringel (see [4]) states that every connected graph,
but K2, is antimagic. Our main result validates this conjecture for graphs having minimum
degree (log n). The proof combines probabilistic arguments with simple tools from analytic
number theory and combinatorial techniques. We also prove that complete partite graphs (but
K2) and graphs with maximum degree at least n  2 are antimagic.
AMS classification code: 05C78
Keywords: Antimagic, Labeling
1 Introduction
All graphs in this paper are finite, undirected and simple. We follow the notation and terminology
of [2]. An antimagic labeling of a graph with m edges and n vertices is a bijection from the set of
