An undirected simple graph G = (V, E) is called antimagic if there exists an injective function f :, the authors gave a proof that regular graphs are antimagic. However, the proof of the main theorem is incorrect as one of the steps uses an invalid assumption. The aim of the present erratum is to fix the proof.

No abstract

One of the most intriguing unsolved questions of matroid optimization is the characterization of the existence of k disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures that can be formulated as special cases, such as Woodall's conjecture on packing disjoint dijoins in a directed graph, or Rota's beautiful conjecture on rearrangements of bases.In the present paper we prove that the problem is difficult under the rank oracle model, i.e., we show that there is no algorithm which decides if the common ground set of two matroids can be partitioned into k common bases by using a polynomial number of independence queries. Our complexity result holds even for the very special case when k = 2.Through a series of reductions, we also show that the abstract problem of packing common bases in two matroids includes the NAE-SAT problem and the Perfect Even Factor problem in directed graphs. These results in turn imply that the problem is not only difficult in the independence oracle model but also includes NP-complete special cases already when k = 2, one of the matroids is a partition matroid, while the other matroid is linear and is given by an explicit representation.

The main result of the paper is motivated by the following two, apparently unrelated graph optimization problems: (A) as an extension of Edmonds' disjoint branchings theorem, characterize digraphs comprising k disjoint branchings B i each having a specified number µ i of arcs, (B) as an extension of Ryser's maximum term rank formula, determine the largest possible matching number of simple bipartite graphs complying with degree-constraints. The solutions to these problems and to their generalizations will be obtained from a new min-max theorem on covering a supermodular function by a simple degree-constrained bipartite graph. A specific feature of the result is that its minimum cost extension is already NP-hard. Therefore classic polyhedral tools themselves definitely cannot be sufficient for solving the problem, even though they make some good service in our approach.

In [18], L. Lovász provided simple and short proofs for two classic minmax theorems of graph theory by inventing basic techniques to handle sub-or supermodular functions. In this paper, we want to demonstrate that these ideas are alive after thirty years of their birth.