Diameter two orientability of mixed graphs

TL;DR

This paper investigates the minimum degree conditions needed for mixed graphs, containing both directed and undirected edges, to be oriented with diameter two, extending prior results from undirected graphs.

Contribution

It establishes the minimum degree threshold for diameter two orientations in mixed graphs, generalizing earlier findings on undirected graphs.

Findings

Determined the minimum degree threshold for mixed graphs.

Extended previous undirected graph results to mixed graphs.

Provided new insights into graph orientation with diameter constraints.

Abstract

In 1967, Katona and Szemer\'{e}di showed that no undirected graph with $n$ vertices and fewer than $\frac{n}{2}\log_2\frac{n}{2}$ edges admits an orientation of diameter two. In 1978, Chv\'atal and Thomassen revealed the complexity of determining whether an undirected graph can be oriented to achieve a diameter of two, proving it to be NP-complete. This breakthrough has sparked ongoing interest in identifying sufficient conditions for graphs to be oriented with the smallest possible diameter of two -- critical for optimizing communication and network flow in larger structures. In 2019, Czabarka, Dankelmann, and Sz\'ekely significantly advanced this field by establishing that the minimum degree threshold for achieving such an orientation in undirected graphs of order $n$ is $\frac{n}{2} + \Theta(\ln n)$. In this paper, we extend this foundational result by determining the minimum degree threshold necessary for realizing an orientation with diameter two in mixed graphs, which contain both undirected and directed edges. Mixed graphs offer a versatile framework, representing an intermediate stage in the orientation process, making our findings a substantial generalization of previous results.