We attempt to generalize a theorem of Nash-Williams stating that graph has k-arc-connected orientation if and only it is 2k-edge-connected. In strongly connected digraph we call an arc deletable its deletion leaves digraph. Given 3-edge-connected G, define Frank number f(G) be the minimum k such there exist orientations G with property every edge becomes in at least one these orientations. are ...

Nash-Williams proved that the edges of a k-edge connected (undirected) graph can be oriented such that the resulting directed graph is ⌊ 2 ⌋-edge connected. A long-standing goal in the area is to obtain analogous results for other types of connectivity, such as node connectivity, element connectivity, and hypergraph edge connectivity. We focus on two special classes of graphs, namely, incidence...

Abstract In an isolate-free graph

A graph G = (V, E) is called weakly four-connected if G is 4-edge-connected and G − x is 2-edge-connected for all x ∈ V . We give sufficient conditions for the existence of ‘splittable’ vertices of degree four in weakly four-connected graphs. By using these results we prove that every minimally weakly fourconnected graph on at least four vertices contains at least three ‘splittable’ vertices of...

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It was proved that every 3-connected bipartite graph admits a vertex-coloring S-edge-weighting for S = {1, 2} (H. Lu, Q. Yu and C. Zhang, Vertex-coloring 2-edge-weighting of graphs, European J. Combin., 32 (2011), 22-27). In this paper, we show that every 2-connected and 3-edge-connected bipartite graph admits a vertex-coloring S-edgeweighting for S ∈ {{0, 1}, {1, 2}}. These bounds we obtain ar...