An orientation of a graph $G$ is { \it in-out-proper} if any two adjacent vertices have different in-out-degrees, where the in-out-degree of each vertex is equal to the in-degree minus the out-degree of that vertex. The { \it in-out-proper orientation number} of a graph $G$, denoted by $\overleftrightarrow{ \chi}(G)$, is $ \min_{ D\in \Gamma}\max_{ v\in V(G)} |d_D^{ \pm}(v)|$, where $\Gamma$ is the set of in-out-proper orientations of $G$ and $d_D^{ \pm}(v)$ is the in-out-degree of the vertex $v$ in the orientation $D$.
Categories: math.CO