Reliable Orientations of Eulerian Graphs

We present a characterization of Eulerian graphs that have a k-arc-connected orientation so that the deletion of any vertex results in a (k− 1)-arc-connected directed graph. This provides an affirmative answer for a conjecture of Frank [2]. The special case, when k = 2, describes Eulerian graphs admitting 2-vertexconnected orientations. This case was proved earlier by Berg and Jordán [1]. These results are specializations of the related results from [5]. This paper concerns orientations of undirected graphs. We denote an undirected graph by G = (V,E) and a directed graph by ~ G = (V,A). Multiple edges are allowed, but loops are not. For an undirected graph G, a set X ⊆ V and u, v ∈ V, let dG(X)= |{xy ∈ E : x ∈ X, y ∈ V −X}|, TG= {v ∈ V : dG(v) is odd}, λG(u, v)= min{dG(Y ) : u ∈ Y, v / ∈ Y }. For a directed graph ~ G, a set X ⊆ V and u, v ∈ V, let δ ~ G(X)= |{xy ∈ A : x ∈ X, y ∈ V − X}|, % ~ G(X)= δ ~ G(V − X), λ ~ G(u, v)= min{δ ~ G(Y ) : u ∈ Y, v / ∈ Y }. An undirected graph G = (V,E) is called k-edge-connected if λG(u, v) ≥ k for all u, v ∈ V. A directed graph D = (V,A) is called k-arc-connected if λ ~ G(u, v) ≥ k for all u, v ∈ V, and D is called k-vertex-connected if |V | > k and G − X is 1arc-connected for all X ⊆ V with |X| < k. Throughout the paper we assume k ≥ 1. The starting point is the weak orientation theorem of Nash-Williams. Theorem 1. [8] A graph G has a k-arc-connected orientation if and only if G is 2k-edge-connected. A pairing M of G is a new graph on vertex set TG in which each vertex has degree one. A pairing M of G is called k-feasible if dM(X) ≤ dG(X)− 2k for all X ⊂ V, X 6= ∅. (1) The following claim is straightforward using the fact, that in any Eulerian orientation ~ G of an Eulerian graph G = (V,E), δ ~ G(X) = % ~ G(X) = dG(X)/2 for all X ⊆ V . Department of Computer Science, and CNL (Communication Networks Lab), Eötvös University, Pázmány Péter sétány 1/C, Budapest, H-1117, Hungary. Research supported by EGRES group (MTA-ELTE) and OTKA grants T 037547 and T046234. Equipe Combinatoire et Optimisation, Université Paris 6, 75252 Paris, Cedex 05, France.