Good acyclic orientations of 4-regular 4-connected graphs

TL;DR

This paper investigates conditions under which 4-regular, 4-connected graphs admit an acyclic orientation with disjoint out- and in-branchings, providing polynomial algorithms and establishing broad classes of graphs with such orientations.

Contribution

It introduces a sufficient condition for good orientations in quartic 2T-graphs, offers polynomial algorithms for recognition and construction, and proves that all 4-regular, 4-connected graphs have good orientations.

Findings

Every 4-regular, 4-connected graph has a good orientation.

Polynomial algorithms are provided for recognizing and constructing good orientations in certain quartic graphs.

Graphs with minimum degree at least half their vertices also have good orientations.

Abstract

We study graphs which admit an acyclic orientation that contains an out-branching and in-branching which are arc-disjoint (such an orientation is called {\bf good}). A {\bf 2T-graph} is a graph whose edge set can be decomposed into two edge-disjoint spanning trees. Clearly a graph has a good orientation if and only if it contains a spanning 2T-graph with a good orientation, implying that 2T-graphs play a central role. Vertex-minimal 2T-graphs with at least two vertices, also known as {\bf generic circuits}, play an important role in rigidity theory for graphs. It was shown in \cite{bangGOpaper} that every generic circuit has a good orientation. Using this, several results on good orientations of 2T-graphs were obtained in \cite{bangGOpaper}. It is an open problem whether there exist a polynomial algorithm for deciding whether a given 2T-graph has a good orientation. In \cite{bangGOpaper} complex constructions of 2T-graphs with no good orientation were given, indicating that the problem might be very difficult. In this paper we focus on so-called {\bf quartics} which are 2T-graphs where every vertex has degree 3 or 4. We identify a sufficient condition for a quartic to have a good orientation, give a polynomial algorithm to recognize quartics satisfying the condition and a polynomial algorithm to produce such an orientation when this condition is met. As a consequence of these results we prove that every 4-regular and 4-connected graph has a good orientation. \iffalse We also provide evidence that even for quartics it may be difficult to find a characterization of those instances which have a good orientation.\fi We also show that every graph on $n\geq 8$ vertices and of minimum degree at least $\lfloor{}n/2\rfloor$ has a good orientation. Finally we pose a number of open problems.