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} $d^*_G(x)$ 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 $\Delta^*(G) = \max\{d^*_G(v) : v \in V(G)\}$. We prove that \begin{align*} \text{(1)}\quad & \overrightarrow{\mathrm{diam}}(G) \leq n - \Delta^* + 3 && \text{if $G$ is undirected, or contains a vertex $u$ with $d^*_G(u) = \Delta^*$} \\ & && \text{and $d^+_G(u) + d^-_G(u) \geq 2$, or $\Delta^* = 5$ and $d^+_G(u) + d^-_G(u) = 1$;} \\ \text{(2)}\quad & \overrightarrow{\mathrm{diam}}(G) \leq n - \Delta^* + 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{diam}}(G) \leq 2(|A| - d(u)) + 7 && \text{if $G$ is undirected;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \text{(2)}\quad & \overrightarrow{\mathrm{diam}}(G) \leq 2(|A| - d^*(u)) + 8 && \text{if $d^+_G(u) + d^-_G(u) \geq 2$;} \\ \text{(3)}\quad & \overrightarrow{\mathrm{diam}}(G) \leq 2(|A| - d^*(u)) + 10 && \text{otherwise}. \end{align*} All of the above bounds are sharp, except possibly the last one.