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.
We consider the following network design problem, that we call the Generalized Terminal Backup Problem: given a graph (or a hypergraph) G 0 = (V, E 0), a set of (at least 2) terminals T ⊆ V and a requirement r(t) for every t ∈ T , nd a multigraph G = (V, E) such that λ G 0 +G (t, T − t) ≥ r(t) for any t ∈ T. In the minimum cost version the objective is to nd G minimizing the total cost c(E) = uv∈E c(uv), given also costs c(uv) ≥ 0 for every pair u, v ∈ V. In the degree-specied version the question is to decide whether such a G exists, satisfying that the number of edges is a prescribed value m(v) at each node v ∈ V. The Terminal Backup Problem solved in [1] is the special case where G 0 is the empty graph and r(t) = 1 for every terminal t ∈ T. We solve the Generalized Terminal Backup Problem in the following two cases. In the rst case we solve the degree-specied version by a splitting-o theorem. This splitting-o theorem in turn provides the solution for the minimum cost version in the case when c is node-induced, that is c(uv) = w(u) + w(v) for some node weights w : V → R +. In the second solved case we turn to the general minimum cost version, and we are able to solve it when G 0 is the empty graph. This includes the Terminal Backup Problem [1] (r ≡ 1) and the Maximum-Weight b-matching Problem (T = V). The solution depends on an interesting new variant of a theorem of Lovász and Cherkassky, and on the solution of the so-called Simplex Matching problem [1]. Our algorithms run in strongly polynomial time for both problems.
In this paper we consider problems related to Nash-Williams´ Strong Orientation Theorem and Odd-Vertex Pairing Theorem. These theorems date to 1960 and up to now not much is known about their relationship to other subjects in graph theory. We investigated many approaches to find a more transparent proof for these theorems and possibly generalizations of them. In many cases we found negative answers: counter-examples and NP-completeness results. For example we show that the weighted and the degree-constrained versions of the well-balanced orientation problem are NP-hard. We also show that it is NP-hard to find a minimum cost feasible odd-vertex pairing or to decide whether two graphs with some common edges have simultaneous well-balanced orientations or not. Nash-Williams´ original approach was to define best-balanced orientations with feasible odd-vertex pairings: we show here that not every best-balanced orientation can be obtained this way. However we prove that in the global case this is true: every smooth k-arc-connected orientation can be obtained through a k-feasible odd-vertex pairing. The aim of this paper is to help to find a transparent proof for the Strong Orientation Theorem. In order to achieve this we propose some other approaches and raise some open questions, too. (c) 2008 Elsevier B.V. All rights reserved
Given a digraph D = (V, A) and a positive integer k, an arc set F ⊆ A is called a karborescence if it is the disjoint union of k spanning arborescences. The problem of finding a minimum cost k-arborescence is known to be polynomial-time solvable using matroid intersection. In this paper we study the following problem: find a minimum cardinality subset of arcs that contains at least one arc from every minimum cost k-arborescence. For k = 1, the problem was solved in [A. Bernáth, G. Pap , Blocking optimal arborescences, IPCO 2013]. In this paper we give an algorithm for general k that has polynomial running time if k is fixed.
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