Diameter Three Orientability of Bipartite Graphs

Bin Chen,An Chang

doi:10.37236/9723

Abstract

In 2019, Czabarka, Dankelmann and Székely showed that for every undirected graph of order $n$, the minimum degree threshold for diameter two orientability is $\frac{n}{2}+ \Theta(\ln n)$. In this paper, we consider bipartite graphs and give a sufficient condition in terms of the minimum degree for such graphs to have oriented diameter three. We in particular prove that for balanced bipartite graphs of order $n$, the minimum degree threshold for diameter three orientability is $\frac{n}{4}+\Theta(\ln n)$.

Highlights

We will use the notation from [1]
The order of G is the number of vertices in G
We denote by δ(G) and ∆(G) the minimum and maximum degrees of the vertices of G

Summary

Introduction

We will use the notation from [1]. All graphs and digraphs considered here have no loops or parallel edges/arcs. In 2018, Dankelmann, Guo and Surmacs [7] considered the oriented diameter of a bridgele−−ss−→graph G of order n in terms of the maximum degree They proved the upper bound diam(G) n − ∆(G) + 3. They constructed an infinite family of undirected graphs whose oriented diameter is equal to this upper bound They showed that, if G is bipartite, the bound above can be improved: Theorem 3. They constructed an infinite family of bridgeless balanced bipartite graphs of order n whose oriented diameter reaches this upper bound. We construct an infinite family of bipartite graphs with oriented diameter at least four, which have almost as big minimum degrees in the partite sets as the bound in our sufficient condition

Main results

Problem