Good acyclic orientations of 4‐regular 4‐connected graphs.

An st $st$‐ordering of a graph G=(V,E) $G=(V,E)$ is an ordering v1,v2,...,vn ${v}_{1},{v}_{2},\ldots ,{v}_{n}$ of its vertex set such that s=v1,t=vn $s={v}_{1},t={v}_{n}$ and every vertex vi ${v}_{i}$ with i=2,3,...,n−1 $i=2,3,\ldots ,n-1$ has both a lower numbered and a higher numbered neighbor. Such orderings have played an important role in algorithms for planarity testing. It is well‐known that every 2‐connected graph has an st $st$‐ordering for every choice of distinct vertices s,t $s,t$. An st $st$‐ordering of a graph G $G$ corresponds directly to a so‐called bipolar orientation of G $G$, that is, an acyclic orientation D $D$ of G $G$ in which s $s$ is the unique source and t $t$ is the unique sink. Clearly every bipolar orientation of a graph has an out‐branching rooted at the source vertex and an in‐branching rooted at the sink vertex. In this paper, we study graphs which admit a bipolar orientation that contains an out‐branching and in‐branching which are arc‐disjoint (such an orientation is called good). A 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. It is a well‐known result due to Tutte and Nash‐Williams, respectively, that every 4‐edge‐connected graph contains a spanning 2T‐graph. Vertex‐minimal 2T‐graphs with at least two vertices, also known as generic circuits, play an important role in rigidity theory for graphs. Recently with Bessy and Huang we proved that every generic circuit has a good orientation. In fact, we may specify the roots of the two branchings arbitrarily as long as they are distinct. Using this, several results on good orientations of 2T‐graphs were obtained. It is an open problem whether there exists a polynomial algorithm for deciding whether a given 2T‐graph has a good orientation. Complex constructions of 2T‐graphs with no good orientation were given in work by Bang‐Jensen, Bessy, Huang and Kriesell (2021) indicating that the problem might be very difficult. In this paper, we focus on so‐called 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 a good 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, where, as for generic circuits, we may specify the roots of the two branchings arbitrarily as long as they are distinct. We also provide evidence that even for quartics it may be difficult to find a characterization of those instances which have a good orientation. We also show that every graph on n≥8 $n\ge 8$ vertices and of minimum degree at least ⌊n∕2⌋ $\lfloor n\unicode{x02215}2\rfloor $ has a good orientation. Finally we pose a number of open problems. [ABSTRACT FROM AUTHOR]