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

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.11585
 Bibcode:
 2021arXiv211011585I
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics;
 Computer Science  Data Structures and Algorithms