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

An oriented graph $\overrightarrow{G}$ is pushably $k$-critical if it is not pushably $k$-colorable, but every proper subgraph of $\overrightarrow{G}$ 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 $\frac{15n+2}{13}$ 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 $\frac{30}{13}$ 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 $\frac{30}{13}$ 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 $K_{2,2,2}$) which (tightly) improves a result due to Borodin \textit{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 $q \leq p$. All these implications improve previously known results.