On arc-density of pushably $3$-critical oriented graphs

An oriented graph G is pushably k-critical if it is not pushably k-colorable, but every proper subgraph of G is. The main result of this article is that every pushably $3$-critical oriented graph on n vertices, but for four exceptions, has at least 15n+2/13 arcs, and that this bound is tight. As an application of this result, we show that the class of oriented graphs with maximum average degree strictly less than 30/13 and girth at least $5$, which includes all oriented planar and projective planar graphs with girth at least $15$, have pushable chromatic number at most $3$. Moreover, we provide an exhaustive list of pushably $3$-critical graphs with maximum average degree equal to 30/13 and a pushably $3$-critical orientation of a $4$-cycle to prove the tightness of our bound with respect to both maximum average degree and girth. We also show that these classes of oriented graphs admit a homomorphism to an oriented planar graph on six vertices (an orientation of K2,2,2) which (tightly) improves a result due to Borodin et al. [Discrete Mathematics 1998]. Furthermore, for these classes of oriented graphs, we prove that the $2$-dipath $L(p,q)$ and the oriented $L(p,q)$ spans are upper bounded by $2p+3q$ for all q ≤ p. All these implications improve previously known results.