Related papers: An Improvement to Chv\'atal and Thomassen's Upper …
Let $f(d)$ be the smallest value for which every bridgeless graph $G$ with diameter $d$ admits a strong orientation $\overrightarrow{G}$ such that the diameter of $\overrightarrow{G}$ is at most $f(d)$. Chv\'atal and Thomassen (1978)…
In 1978, Chv\'atal and Thomassen showed that every bridgeless graph with diameter 2 has an orientation with diameter at most 6. They also gave general bounds on the smallest value $f(d)$ such that every bridgeless graph $G$ with diameter…
Erd\H{o}s, Pach, Pollack, and Tuza [\textit{J. Combin. Theory Ser. B, 47(1) (1989), 73-79}] proved that the diameter of a connected $n$-vertex graph with minimum degree $\delta$ is at most $\frac{3n}{\delta+1}+O(1)$. The oriented diameter…
Let $G$ be a connected bridgeless graph with domination number $\gamma$. The oriented diameter (strong diameter) of $G$ is the smallest integer $d$ for which $G$ admits a strong orientation with diameter (strong diameter) $d$. Kurz and…
The diameter of an undirected or a directed graph is defined to be the maximum shortest path distance over all pairs of vertices in the graph. Given an undirected graph $G$, we examine the problem of assigning directions to each edge of $G$…
Let $f(d)$ be the smallest value for which every bridgeless graph $G$ with diameter $d$ admits a strong orientation $\overrightarrow{G}$ such that the diameter of $\overrightarrow{G}$ is at most $f(d)$. Chv\'atal and Thomassen (JCT-B, 1978)…
A strong orientation of a graph $G$ is an assignment of a direction to each edge such that $G$ is strongly connected. The oriented diameter of $G$ is the smallest diameter among all strong orientations of $G$. A block of $G$ is a maximal…
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…
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…
A mixed multigraph is a multigraph which may contain both undirected and directed edges. An orientation of a mixed multigraph $G$ is an assignment of exactly one direction to each undirected edge of $G$. A mixed multigraph $G$ can be…
A proper orientation $D$ of an undirected graph $G$ is an orientation of $G$ such that $d_D^+(u)\not=d_D^+(v)$ for any edge $uv\in E(G)$. Denote the proper orientation number $\vec{\chi}(G)$ of an undirected graph $G$ as the minimum…
In this paper, we show that the oriented diameter of any $n$-vertex $2$-connected near triangulation is at most $\lceil{\frac{n}{2}}\rceil$ (except for seven small exceptions), and the upper bound is tight. This extends a result of Wang…
The oriented diameter of a bridgeless graph $G$ is $\min\{diam(H)\ | H\ is\ an orientation\ of\ G\}$. A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called rainbow if no two edges of the path are…
For a positive integer $r$, let $f(r)$ denote the smallest number such that any 2-edge connected mixed graph with radius $r$ has an oriented radius of at most $f(r)$. Recently, Babu, Benson, and Rajendraprasad significantly improved the…
The diameter of a directed graph is the maximum distance between any pair of vertices. We study a problem that generalizes \textsc{Oriented Diameter}: For a given directed graph and a positive integer $d$, what is the minimum number of arc…
An orientation $D$ of a graph $G=(V,E)$ is a digraph obtained from $G$ by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each $v \in V(G)$, the indegree of $v$ in $D$, denoted by $d^-_D(v)$, is…
Computing the diameter of a graph, i.e. the largest distance, is a fundamental problem that is central in fine-grained complexity. In undirected graphs, the Strong Exponential Time Hypothesis (SETH) yields a lower bound on the time vs.…
In 2015, Dankelmann and Bau proved that for every bridgeless graph $G$ of order $n$ and minimum degree $\delta$ there is an orientation of diameter at most $11\frac{n}{\delta+1}+9$. In 2016, Surmacs reduced this bound to…
An oriented graph has weak diameter at most $d$ if every non-adjacent pair of vertices are connected by a directed $d$-path. The function $f_d(n)$ denotes the minimum number of arcs in an oriented graph on $n$ vertices having weak diameter…
An oriented graph is a directed graph with no bi-directed edges, i.e. if $xy$ is an edge then $yx$ is not an edge. The oriented size Ramsey number of an oriented graph $H$, denoted by $r(H)$, is the minimum $m$ for which there exists an…