Monotone edge flips to an orientation of maximum edge-connectivity à la Nash-Williams
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We initiate the study of k-edge-connected orientations of undirected graphs through edge flips for k ≥ 2. We prove that in every orientation of an undirected 2k-edge-connected graph, there exists a sequence of edges such that flipping their directions one by one does not decrease the edge-connectivity, and the final orientation is k-edge-connected. This yields an “edge-flip based” new proof of Nash-Williams' theorem: an undirected graph G has a k-edge-connected orientation if and only if G is 2k-edge-connected. As another consequence of the theorem, we prove that the edge-flip graph of k-edge-connected orientations of an undirected graph G is connected if G is (2k+2)-edge-connected. This has been known to be true only when k=1.
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