New results on proper orientation number of graphs

The proper orientation number $\vec{\chi }(G)$ of an undirected graph $G$ is the minimum $k$ such that there exists an orientation of $G$ with all out-degrees at most $k$ and with different out-degrees for any two adjacent vertices. Chen, Mohar and Wu (JCTB, 2023) proved that if $G$ is a $r$-partite graph, then $\vec{\chi }(G)\le \frac{1}{2}\text{Mad}(G)+r^{1+o(1)}$, where $\text{Mad}(G)$ is the maximum average degree of $G$. Moreover, if $G$ is a bipartite graph, then $\vec{\chi }(G)\le \lceil \frac{1}{2}\text{Mad}(G)\rceil +3$ and this bound is tight. They also asked whether $\vec{\chi }(G)-\lceil \frac{1}{2}\text{Mad}(G)\rceil$ can be bounded by a linear function of $r$. In this paper, we first construct somewhat involved $r$-partite graphs with $\vec{\chi }(G)\ge \lceil \frac{1}{2}\text{Mad}(G)\rceil +\lfloor \frac{5}{2}r\rfloor -2$, showing that a linear dependence on $r$ is unavoidable. We also prove that $\vec{\chi }(G)\le \lceil \frac{1}{2}\text{Mad}(G)\rceil +7$ for every 3-partite graph $G$. This implies $\vec{\chi }(G)\le 10$ for $3$-colorable planar graphs and $\vec{\chi }(G)\le 9$ for outerplanar graphs, improving the corresponding bounds of Chen, Mohar, and Wu.