The oriented diameter of a bridgeless graph with the given path Pk

Abstract

Let $G=(V,E)$ be a bridgeless undirected graph. The oriented diameter of G is the minimum diameter of any strongly connected orientation of G. Dankelmann, Guo and Surmacs [J. Graph Theory, 88 (2018), 5-17] showed that every bridgeless graph G of order n has an oriented diameter at most $n-\Delta +3$, where Δ is the maximum degree of G. By defining $N_G\left(H\right)={\bigcup }_{v\in V\left(H\right)}{N}_G\left(v\right)\setminus V\left(H\right)$ for every subgraph H of G, they proved that for an edge e, G has an orientation of diameter at most $n-\left|N_G\right(e\left)\right|+5$. In this paper, we generalize the above-mentioned results by substituting a vertex or an edge e by a given path $P_k=v_1{v}_2\cdots v_k$ in G. We first give an algorithm to cover $P_k$ with some specific cycles, and then prove the upper bound $n-\left|N_G\right(P_k\left)\right|+2\lfloor \frac{k}{2}\rfloor +3$ on the oriented diameter. We provide examples to show that our bound is sharp.