Dense graphs are antimagic

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 K 2 , 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 K 2 ) and graphs with maximum degree at least n À 2 are antimagic.

On antimagic directed graphs

“…Apart from Theorem 1.1, our results in this setting are fairly modest. In the undirected case, it is straightforward to prove that any graph on n vertices with a vertex of degree n−1 is antimagic (even a vertex of degree n−2 suffices [1], though this is no longer trivial). Proving that every orientation of such a graph is antimagic, however, seems rather difficult.…”

“…Starting with the former question, it is possible to adapt the proof of Alon et al [1] of the aforementioned result to directed graphs (though this is not entirely straightforward as some technical problems arise). This provides an affirmative answer to the first question for "dense" graphs.…”

“…This conjecture is still open; however, some special cases of it were proved. Alon et al [1] proved that large dense graphs are antimagic; that is, there exists an absolute constant C>0 such that any graph with n vertices and minimum degree at least C log n is antimagic. Hefetz [7] proved that any graph on n = 3 k vertices that admits a triangle factor is antimagic.…”

Approximate results for rainbow labelings

“…up to a sign (see e.g. Alon [3].) We next show an analogous result for the fourth power of the Vandermonde polynomial.…”

“…While in general the Hartsfield and Ringel conjecture remains open, some partial results are known which support the conjecture. Alon et al [3] used probabilistic methods and some techniques from analytic number theory to show that the conjecture is true for all graphs having minimum degree at least Ω(log |V (G)|). They also proved that if G is a graph with order |V (G)| ≥ 4 and maximum degree ∆(G),…”