Monotone edge flips to an orientation of maximum edge-connectivity \`a la Nash-Williams

TL;DR

This paper introduces a method using edge flips to transform undirected graphs into orientations with maximum edge-connectivity, providing new proofs and extending known connectivity results for these orientations.

Contribution

It establishes that any orientation of a 2k-edge-connected graph can be reached through edge flips to a k-edge-connected orientation, offering a new proof of Nash-Williams' theorem and extending connectivity results.

Findings

Existence of edge-flip sequences preserving edge-connectivity

New proof of Nash-Williams' theorem using edge flips

Connectivity of the edge-flip graph for higher edge-connectivity

Abstract

We initiate the study of $k$-edge-connected orientations of undirected graphs through edge flips for $k \geq 2$. 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 $k=1$.