Oriented Diameter of Mixed Graphs with Given Maximum Undirected Degree

In 2018, Dankelmann, Gao, and Surmacs [J. Graph Theory, 88(1): 5–17, 2018] established sharp bounds on the oriented diameter of a bridgeless undirected graph and a bridgeless undirected bipartite graph in terms of vertex degree. In this paper, we extend these results to \emph{mixed graphs}, which contain both directed and undirected edges. Let the \emph{undirected degree} $d_G^{\ast }(x)$ of a vertex $x\in V(G)$ be the number of its incident undirected edges in a mixed graph $G$ of order $n$, and let the \emph{maximum undirected degree} be ${\mathrm{\Delta }}^{\ast }(G)=\max \{d_G^{\ast }(v):v\in V(G)\}$. We prove that \begin{align*} $\text{(1)}$\quad & \overrightarrow{$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}$}(G) \leq n - Δ^* + 3 && $\text{if }G\text{ is undirected, or contains a vertex }u\text{ with }{d}_G^{\ast }(u)={\mathrm{\Delta }}^{\ast }$ \ & && $\text{and }{d}_G^{+}(u)+d_G^{-}(u)\ge 2\text{, or }{\mathrm{\Delta }}^{\ast }=5\text{ and }{d}_G^{+}(u)+d_G^{-}(u)=1\text{;}$ \ $\text{(2)}$\quad & \overrightarrow{$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}$}(G) \leq n - Δ^* + 4 && $\text{otherwise}$. \end{align*} We also establish bounds for mixed bipartite graphs. If $G$ is a bridgeless mixed bipartite graph with partite sets $A$ and $B$, and $u\in B$, then \begin{align*} $\text{(1)}$\quad & \overrightarrow{$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}$}(G) \leq 2(|A| - d(u)) + 7 && $\text{if }G\text{ is undirected; }$ \ $\text{(2)}$\quad & \overrightarrow{$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}$}(G) \leq 2(|A| - d^(u)) + 8 && $\text{if }{d}_G^{+}(u)+d_G^{-}(u)\ge 2\text{;}$ \ $\text{(3)}$\quad & \overrightarrow{$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}$}(G) \leq 2(|A| - d^(u)) + 10 && $\text{otherwise}$. \end{align*} All of the above bounds are sharp, except possibly the last one.