A vertex set S ⊆ V (G) is k-weak-edge-connected if for every C ⊂ S and x ∈ S−C there are min{k, |C|} edge-disjoint (x,C)-paths in G. For a graph G and a k-weakedge-connected vertex set S ⊂ V (G) with k ≥ 3 and 4 ≤ |S| ≤ 2k, we show that G has an eulerian subgraph containing all vertices in S.

If G is a 2-connected bipartite biclaw-free graph with (G) 4, then by [2, Lemma 2.2], every edge of G lies in a 4-cycle, and then by Lemma 2.5 (the correct version), G is collapsible. It follows that G will have a spanning eulerian subgraph. Note that a Hamiltonian cycle of G is a spanning eulerian subgraph of G with maximum degree 2. We consider it one possible way to attack Conjecture 1.1 of ...

We present an efficient distributed algorithm for constructing an Eulerian tour in a network. To construct an Eulerian circuit the algorithm requires (1 + r)(IEI + IVl) messages and time units, where IEI is the number of the communication links, IVl is the number of the nodes in the underlying network graph, and 0 5 r < 1 . The value of r depends on the network topology and on the chosen traver...

An Eulerian subgraph in a digraph is a subgraph in which indegree equals outdegree at each vertex. Alon and Tarsi [2] proved that a graph is d-choosable when it has an orientation that has no vertex of outdegree at least d and has the property that the numbers of Eulerian subgraphs with even-sized and odd-sized edge sets differ. We generalize this theorem to k-uniform hypergraphs, where k is pr...

We describe and analyze a new construction that produces new Eulerian lattices from old ones. It specializes to a construction that produces new strongly regular cellular spheres (whose face lattices are Eulerian). The construction does not always specialize to convex polytopes; however, in a number of cases where we can realize it, it produces interesting classes of polytopes. Thus we produce ...

The notion of level posets is introduced. This class of infinite posets has the property that between every two adjacent ranks the same bipartite graph occurs. When the adjacency matrix is indecomposable, we determine the length of the longest interval one needs to check to verify Eulerianness. Furthermore, we show that every level Eulerian poset associated to an indecomposable matrix has even ...

We consider the multiflow feasibility problem whose demand graph is the vertex-disjoint union of two triangles. We show that this problem has a 1/12-integral solution or no solution under the Euler condition. This solves a conjecture raised by Karzanov, and completes the classification of the demand graphs having bounded fractionality. We reduce this problem to the multiflow maximization proble...

We consider the multiflow feasibility problem whose demand graph is the vertexdisjoint union of two triangles. We show that this problem has a 1/12-integral solution whenever it is feasible and satisfies the Euler condition. This solves a conjecture raised by Karzanov, and completes the classification of the demand graphs having bounded fractionality. We reduce this problem to the multiflow max...

We prove a decomposition result for locally finite graphs which can be used to extend results on edge-connectivity from finite to infinite graphs. It implies that every 4k-edge-connected graph G contains an immersion of some finite 2k-edge-connected Eulerian graph containing any prescribed vertex set (while planar graphs show that G need not contain a subdivision of a simple finite graph of lar...

A colored eulerian digraph is an eulerian digraph G where a color is assigned to the tail of each edge and a color is assigned to the head of each edge. A compatible circuit is an eulerian circuit such that for every two consecutive edges uv and vw of the circuit, the color of the head of uv is different from the color of the tail of vw. Let S3 be the set of vertices of outdegree and indegree t...