Connectivity of orientations of 3-edge-connected graphs

12/06/2020
by   Florian Hörsch, et al.
0

We attempt to generalize a theorem of Nash-Williams stating that a graph has a k-arc-connected orientation if and only if it is 2k-edge-connected. In a strongly connected digraph we call an arc deletable if its deletion leaves a strongly connected digraph. Given a 3-edge-connected graph G, we define its Frank number f(G) to be the minimum number k such that there exist k orientations of G with the property that every edge becomes a deletable arc in at least one of these orientations. We are interested in finding a good upper bound for the Frank number. We prove that f(G)≤ 7 for every 3-edge-connected graph. On the other hand, we show that a Frank number of 3 is attained by the Petersen graph. Further, we prove better upper bounds for more restricted classes of graphs and establish a connection to the Berge-Fulkerson conjecture. We also show that deciding whether all edges of a given subset can become deletable in one orientation is NP-complete.

READ FULL TEXT
research
10/22/2021

Monotone edge flips to an orientation of maximum edge-connectivity à la Nash-Williams

We initiate the study of k-edge-connected orientations of undirected gra...
research
05/03/2023

Frank number and nowhere-zero flows on graphs

An edge e of a graph G is called deletable for some orientation o if the...
research
12/14/2021

A note on 2-vertex-connected orientations

We consider two possible extensions of a theorem of Thomassen characteri...
research
06/03/2023

Make a graph singly connected by edge orientations

A directed graph D is singly connected if for every ordered pair of vert...
research
04/28/2023

Directed hypergraph connectivity augmentation by hyperarc reorientations

The orientation theorem of Nash-Williams states that an undirected graph...
research
09/13/2022

Structure and Complexity of Graphical Designs for Weighted Graphs through Eigenpolytopes

We extend the theory of graphical designs, which are quadrature rules fo...
research
11/30/2020

On the proper orientation number of chordal graphs

An orientation D of a graph G=(V,E) is a digraph obtained from G by repl...

Please sign up or login with your details

Forgot password? Click here to reset