Oriented Diameter of Mixed Graphs with Given Maximum Undirected Degree

TL;DR

This paper extends bounds on the oriented diameter from undirected graphs to mixed graphs with both directed and undirected edges, providing sharp bounds based on maximum undirected degree and bipartite structure.

Contribution

It introduces new sharp bounds on the oriented diameter of mixed graphs and bipartite graphs, generalizing previous results to more complex graph structures.

Findings

Bounds on oriented diameter for mixed graphs are established.

Bounds for mixed bipartite graphs are derived.

Most bounds are proven to be sharp.

Abstract

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.