New results on proper orientation number of graphs

A proper orientation $D$ of an undirected graph $G$ is an orientation of $G$ such that $d_D^+(u)\not=d_D^+(v)$ for any edge $uv\in E(G)$. Denote the proper orientation number $\vec{\chi}(G)$ of an undirected graph $G$ as the minimum $\Delta^+(D)$ among all proper orientations $D$ of $G$. Chen, Mohar and Wu (JCTB, 2023) proved that if $G$ is a $r$-partite graph, then $\vec{\chi}(G) \leq \frac{1}{2} \text{Mad}(G)+O(\frac{r\log{r}}{\log{\log{r}}})$, where $\text{Mad}(G)$ is the maximum average degree of $G$. Moreover, if $G$ is a bipartite graph, then $\vec{\chi}(G) \leq \lceil \frac{1}{2} \text{Mad}(G)\rceil +3$, 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 prove that $\vec{\chi}(G) \leq\lceil \frac{1}{2} \text{Mad}(G)\rceil +7$ for every 3-partite graph $G$. As a corollary, we also improve Chen, Mohar and Wu's bounds for the 3-colorable planar graphs and the outerplanar graphs. Our proof use the notion of potential out-degree and weighted matching lemma with special weighted functions. We also construct a class of $r$-partite graphs with $\vec{\chi}(G)\geq\lceil \frac{1}{2} \text{Mad}(G)\rceil +r+1$ to be the possible extremal graphs.