An Orientability of a Graph

Abstract

In 1962 Ore posed the problem, characterize graphs which are cover graphs; i.e., characterize those graphs which are orientable as an ordered set. In 1985 O. Pretzel have studied the orientability of graph in terms of girth and chromatic numbers of a graph. In 1995 Ne\v{s}et\v{r}il and R\"{o}dl have studied the graphs of arbitrary large girth that are not covering graphs. In 2021 Bhavale and Waphare introduced the concept of a poset dismantlable by doubly irreducibles and proved that a graph is orientable as a poset dismantlable by doubly irreducible if and only if it is (non-trivial) adjunct of ears. In this paper, we survey an orientability of several classes of graphs in particular, covering graphs of posets. We will also discuss in detail about the various types of orientability of graphs. In the end, we provide a list of some open problems related to the orientability of graphs.