Oriented Diameter of Mixed Graphs with Given Maximum Undirected Degree
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866911036175548416 |
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| author | An, Ran Li, Hengzhe Liu, Jianbing Sun, Gaoxing |
| author_facet | An, Ran Li, Hengzhe Liu, Jianbing Sun, Gaoxing |
| contents | 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 $Δ^*(G) = \max\{d^*_G(v) : v \in V(G)\}$. We prove that \begin{align*} \text{(1)}\quad & \overrightarrow{\mathrm{diam}}(G) \leq n - Δ^* + 3 && \text{if $G$ is undirected, or contains a vertex $u$ with $d^*_G(u) = Δ^*$} \\ & && \text{and $d^+_G(u) + d^-_G(u) \geq 2$, or $Δ^* = 5$ and $d^+_G(u) + d^-_G(u) = 1$;} \\ \text{(2)}\quad & \overrightarrow{\mathrm{diam}}(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{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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02277 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Oriented Diameter of Mixed Graphs with Given Maximum Undirected Degree An, Ran Li, Hengzhe Liu, Jianbing Sun, Gaoxing Combinatorics 05C07, 05C12, 05C20 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 $Δ^*(G) = \max\{d^*_G(v) : v \in V(G)\}$. We prove that \begin{align*} \text{(1)}\quad & \overrightarrow{\mathrm{diam}}(G) \leq n - Δ^* + 3 && \text{if $G$ is undirected, or contains a vertex $u$ with $d^*_G(u) = Δ^*$} \\ & && \text{and $d^+_G(u) + d^-_G(u) \geq 2$, or $Δ^* = 5$ and $d^+_G(u) + d^-_G(u) = 1$;} \\ \text{(2)}\quad & \overrightarrow{\mathrm{diam}}(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{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. |
| title | Oriented Diameter of Mixed Graphs with Given Maximum Undirected Degree |
| topic | Combinatorics 05C07, 05C12, 05C20 |
| url | https://arxiv.org/abs/2507.02277 |