Sascha Kurz, Martin Laetsch
We consider the problem of finding an orientation with minimum diameter of a connected bridgeless graph. Fomin et. al. discovered a relation between the minimum oriented diameter an the size of a minimal dominating set. We improve their upper bound.
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In 1978, Chvátal 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 $d$ has an orientation with diameter at most $f(d)$. For $d=3$, they proved that $8\leq f(d)\leq 24$. Until recently, Kwok, Liu and West improved the above bounds by proving $9\leq f(3)\leq 11$ in [P.K. Kwok, Q. Liu and D.B. West, Oriented diameter of graphs with diameter 3, J. Combin. Theory Ser.B 100(2010), 265-274]. In this paper, we determine the oriented di…
Oriented matroids (often called order types) are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid. We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the la…
On the oriented diameter of near planar triangulations
本文研究近平面三角剖分的有向直径，证明除七个小例外外，$n$ 顶点 $2$-连通近三角剖分的有向直径不超过 $\lceil n/2\rceil$ 且界紧，改进并推广了已有结果。
Oriented diameter of graphs with given domination number
本文通过收缩不可避免的可选子图并对支配数作归纳，证实了 Kurz 和 Lätsch 关于无桥图有向直径至多为 ⌈(7γ+1)/2⌉ 的猜想，并进一步证明有向强直径至多为 7γ−1 且该界最优。
Oriented diameter of the complete tripartite graph (II)
本文确定了完全三部图K(3,3,q)和K(3,4,q)的有向直径，为求解一般情形K(3,p,q)的有向直径提供了特殊情形结果。
Oriented diameter of the complete tripartite graph (III)
该研究针对无桥图的有向直径问题，确定了$p\geqslant 5$时完全三部图$K(3,p,q)$的有向直径，结合已有成果补全了该类图有向直径的结论。
Let $S(G^σ)$ be the skew-adjacency matrix of the oriented graph $G^σ$, which is obtained from a simple undirected graph $G$ by assigning an orientation $σ$ to each of its edges. The skew energy of an oriented graph $G^σ$ is defined as the sum of absolute values of all eigenvalues of $S(G^σ)$. For any positive integer $d$ with $3\leq d\leq n-3$, we determine the graph with minimal skew energy among all oriented bicyclic graphs that contain no vertex disjoint odd cycle of lengths $s$ and $l$ with $s+l\equiv 2(mod 4)$ on $n$ vertices with a given diameter $d$.
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 colored the same. The rainbow connection number $rc(G)$ of $G$ is the smallest integer $k$ for which there exists a $k$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. In this paper, we obtain upper bounds for the oriented diameter and the rainbow connection number of a graph in terms of $rad(G)$ and $η(G)$, wher…
The asymptotic maximum oriented diameter of graphs
本文证明了连通无桥图的最大定向直径$f(d)$满足$f(d)\le \tfrac12d^2+7d$，匹配已知下界的前导系数，确立了最优二次系数，并给出多项式时间算法构造满足界的强定向。
Improved bounds on the oriented diameter of planar triangulations
本文针对平面三角剖分的定向直径问题，改进了通用上界的首项常数，并给出4-连通平面三角剖分的定向直径新上界。
An improved upper bound on the oriented diameter of graphs with diameter $4$
本文改进了图直径为4时有向直径上界，将最佳上界从21降低至18。
In 2015, Dankelmann and Bau proved that for every bridgeless graph $G$ of order $n$ and minimum degree $δ$ there is an orientation of diameter at most $11\frac{n}{δ+1}+9$. In 2016, Surmacs reduced this bound to $7\frac{n}{δ+1}.$ In this paper, we consider the girth of a graph $g$ and show that for any $\varepsilon>0$ there is a bound of the form $(2g+\varepsilon)\frac{n}{h(δ,g)}+O(1)$, where $h(δ,g)$ is a polynomial. Letting $g=3$ and $\varepsilon<1$ gives an inprovement on the result by Surmacs.
{\em Arrowhead} and {\em diamond} are two hierarchical Cayley graphs defined on the triangular grid. In their undirected version, they are isomorphic and merely define two distinct representations of the same graph. This paper gives the expression of their diameter, in the oriented and non--oriented case. It also displays the full distribution of antipodals.
On Oriented Diameter of Power Graphs
本文研究了群幂图的有向直径，证明了2-边连通幂图有向直径至多为4，循环群幂图有向直径多为2，并给出了非循环幂零群有向直径为3或4的充要条件及多项式时间算法。
An improved upper bound for oriented diameter of graphs with diameter $4$
本文提出序贯定向框架与势函数分析，将直径为4的无桥图的有向直径上界从18改进至16，方法可推广至更大直径情形。
The harmonic index of a graph $G$, is defined as the sum of weights $\frac{2}{d(u)+d(v)}$ of all edges $uv$ of $G$, where $d(u)$ is the degree of the vertex $u$ in $G$. In this paper we find the minimum harmonic index of bicyclic graph of order $n$ and diameter $d$. We also characterized all bicyclic graphs reaching the minimum bound.
Cycles of Length 4 or 8 in Graphs with Diameter 2 and Minimum Degree at Least 3
本文证明了直径为2且最小度数至少为3的图中必存在长度为4或8的环，从而支持了埃尔多斯-加法斯猜想。
An orientation of an undirected graph $G$ is an assignment of exactly one direction to each edge of $G$. The oriented diameter of a graph $G$ is the smallest diameter among all the orientations of $G$. The maximum oriented diameter of a family of graphs $\mathscr{F}$ is the maximum oriented diameter among all the graphs in $\mathscr{F}$. Chvátal and Thomassen [JCTB, 1978] gave a lower bound of $\frac{1}{2}d^2+d$ and an upper bound of $2d^2+2d$ for the maximum oriented diameter of the family of $2$-edge connected graphs of diameter $d$. We improve this upper bound to $ 1.373 d^2 + 6.971d-1 $, w…
On the oriented diameter of graphs with given minimum degree
本文研究给定最小度图的有向直径上界问题，证明使有向直径满足 c·3n/(δ+1)+O(1) 的最小常数 c 为 1，解决了 Bau 和 Dankelmann 提出的最优常数问题。
An asymptotically tight bound on oriented diameter
本文证明了有向直径函数 $f(d)$ 的上界为 $\lceil d^2/2\rceil+6d$，从而确定了其二次项主导系数，并给出渐近紧界。
Diameter of orientations of graphs with given order and number of blocks
本研究针对给定阶数和块数的无桥图，推导得出定向直径的紧上界，并进一步给出无桥块图定向直径分奇偶情形的上界结果。
Geometrical objects with integral sides have attracted mathematicians for ages. For example, the problem to prove or to disprove the existence of a perfect box, that is, a rectangular parallelepiped with all edges, face diagonals and space diagonals of integer lengths, remains open. More generally an integral point set $\mathcal{P}$ is a set of $n$ points in the $m$-dimensional Euclidean space $\mathbb{E}^m$ with pairwise integral distances where the largest occurring distance is called its diameter. From the combinatorial point of view there is a natural interest in the determination of the s…
We show that every sufficiently large oriented graph with minimum in- and outdegree at least (3n-4)/8 contains a Hamilton cycle. This is best possible and solves a problem of Thomassen from 1979.
Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets $\mathcal{P}$, which are sets of $n$ points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the minimum possible diameter $d(2,n)$ of a plane integral point set consisting of $n$ points arises. We give some new exact values and describe state-of-the-art algorithms to obtain them. It turns out that plane integral point sets with minimum diameter c…
Diameter reduction via arc reversal
本文研究通过最少弧反转次数将有向图直径减小至给定上限的问题，证明了特定参数下的困难性，并为平面图和仙人掌图提供了多项式时间算法。
Let $\vec{T}_k$ be the transitive tournament on $k$ vertices. We show that every oriented graph on $n=4m$ vertices with minimum total degree $(11/12+o(1))n$ can be partitioned into vertex disjoint $\vec{T}_4$'s, and this bound is asymptotically tight. We also improve the best known bound on the minimum total degree for partitioning oriented graphs into vertex disjoint $\vec{T}_k$'s.
Counting the minimum number of arcs in an oriented graph having weak diameter 2
本研究针对弱直径为2的n顶点有向图最小弧数f_2(n)的求解难题，通过关联绝对有向团证明其严格递增，改进了已知上界并猜想其为紧界，同时优化了有向消色数的相关界。
Paths with two blocks in oriented graphs of large minimum semi-degree
本文证明了Stein猜想在具有两个块的有向路径情况下的成立，即大最小半度数的有向图包含所有此类路径。
An asymptotically tight bound on oriented diameter
本文证明了有向直径函数$f(d)$满足$f(d)=\lceil d^2/2\rceil+d+O(1)$，通过构造中心子图并联合估计路径，将上下界差缩小到常数。
The minimum size and maximum diameter of an edge-pancyclic graph of a given order
本文研究了给定阶数边泛圈图的最小边数 $f(n)$，给出了其上下界及最大直径，并在3-连通情形下确定了 $f(n)$ 的精确值，同时确定了给定连通性条件下每条边均位于三角形中的图的最小边数。
Embracing exchange sequences and oriented matroid polyhedron diameter
将定向拟阵基的拥抱交换距离归约为定向拟阵多面体的度量，反驳了Caoduro等人和Bérczi与Nádor的近期猜想，同时证明了在秩为r的定向拟阵中任意两个拥抱基可在至多2r^{log_2(r)+3}步内变换，在Lawrence定向拟阵中可在至多r步内变换。
The Steiner $k$-diameter $sdiam_k(G)$ of a graph $G$, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical diameter. When $k=2$, $sdiam_2(G)=diam(G)$ is the classical diameter. The problem of determining the minimum size of a graph of order $n$ whose diameter is at most $d$ and whose maximum is $\ell$ was first introduced by Erdös and Rényi. In this paper, we generalize the above problem for Steiner $k$-diameter, and study the problem of determining the minimum size of a graph of order $n$ whose Steiner $3$-diameter is at most $d$ a…
Oriented diameter of graphs with diameter $4$ and given edge girth
研究无桥图强定向后的有向直径上界，通过边围长参数化，得到直径为4时的若干上界结果。
The Steiner diameter $sdiam_k(G)$ of a graph $G$, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical diameter. When $k=2$, $sdiam_2(G)=diam(G)$ is the classical diameter. The problem of determining the minimum size of a graph of order $n$ whose diameter is at most $d$ and whose maximum is $\ell$ was first introduced by Erdös and Rényi. Recently, Mao considered the problem of determining the minimum size of a graph of order $n$ whose Steiner $k$-diameter is at most $d$ and whose maximum is at most $\ell$, where $3\leq k\leq n$, and…
Erdős, Pach, Pollack and Tuza [J. Combin. Theory, B 47, (1989), 279-285] conjectured that the diameter of a $K_{2r}$-free connected graph of order $n$ and minimum degree $δ\geq 2$ is at most $\frac{2(r-1)(3r+2)}{(2r^2-1)}\cdot \frac{n}δ + O(1)$ for every $r\ge 2$, if $δ$ is a multiple of $(r-1)(3r+2)$. For every $r>1$ and $δ\ge 2(r-1)$, we create $K_{2r}$-free graphs with minimum degree $δ$ and diameter $\frac{(6r-5)n}{(2r-1)δ+2r-3}+O(1)$, which are counterexamples to the conjecture for every $r>1$ and $δ>2(r-1)(3r+2)(2r-3)$. The rest of the paper proves positive results under a stronger hypot…
Oriented paths with two blocks in bipartite oriented graphs
本文证明最小半度至少为 3k/8+2 的二分有向图包含所有长度为 k 的两段有向路径，并指出二分情形下阈值与段数密切相关。
Minimum $k$-Section denotes the NP-hard problem to partition the vertex set of a graph into $k$ sets of sizes as equal as possible while minimizing the cut width, which is the number of edges between these sets. When $k$ is an input parameter and $n$ denotes the number of vertices, it is NP-hard to approximate the width of a minimum $k$-section within a factor of $n^c$ for any $c<1$, even when restricted to trees with constant diameter. Here, we show that every tree $T$ allows a $k$-section of width at most $(k-1) (2 + 16n / diam(T) ) Δ(T)$. This implies a polynomial-time constant-factor appro…
Diameter two orientability of mixed graphs
本文确定了混合图（同时含无向边和有向边）实现直径为二的定向所需的最小度阈值，将无向图的已知结果推广至混合图，为定向优化提供更普适的充分条件。
On the minimum number of eigenvalues of trees of diameter seven
本文研究直径为七的树的最小不同特征值个数，基于种子构造实现，并修正了Johnson和Saiago论文中的一个结果。
For $n \in \mathbb{N}$ let $δ_n$ be the smallest value such that every graph of order $n$ and minimum degree at least $δ_n$ admits an orientation of diameter two. We show that $δ_n=\frac{n}{2} + Θ(\ln n)$.
An oriented k-uniform hypergraph (a family of ordered k-sets) has the ordering property (or Property O) if for every linear order of the vertex set, there is some edge oriented consistently with the linear order. We find bounds on the minimum number of edges in a hypergraph with Property O.
A particular case of Caccetta-Häggkvist conjecture, says that a digraph of order $n$ with minimum out-degree at least $1/3n$ contains a directed cycle of length at most 3. Recently, Kral, Hladky and Norine proved that a digraph of order $n$ with minimum out-degree at least $0.3465n$ contains a directed cycle of length at most 3 (which currently is the best result). A weaker particular case says that a digraph of order $n$ with minimum semi-degree at least $1/3n$ contains a directed triangle. In a recent paper, by using the result of Kral et al, the author proved that for $β\geq 0.343545$, any …
Oriented trees and paths in digraphs
本文综述了有向图中包含所有给定长度定向路径或给定大小定向树的条件，涵盖竞赛图及色数、边密度、最小出度和最小半度等条件，并指出开放问题。
An {\em orientation} of an undirected graph $G$ is an assignment of exactly one direction to each edge of $G$. Converting two-way traffic networks to one-way traffic networks and bidirectional communication networks to unidirectional communication networks are practical instances of graph orientations. In these contexts minimising the diameter of the resulting oriented graph is of prime interest. The $n$-star network topology was proposed as an alternative to the hypercube network topology for multiprocessor systems by Akers and Krishnamurthy [IEEE Trans. on Computers (1989)]. The $n$-star gra…
Powers of Hamilton cycles in oriented and directed graphs
本文研究有向图与定向图中 Hamilton 圈幂的度阈值，渐近确定有向图中 Hamilton 圈平方的最小总度阈值，并给出定向图中保证 k 次幂的半度条件。
Kelly, Kuehn and Osthus conjectured that for any l>3 and the smallest number k>2 that does not divide l, any large enough oriented graph G with minimum indegree and minimum outdegree at least \lfloor |V(G)|/k\rfloor +1 contains a directed cycle of length l. We prove this conjecture asymptotically for the case when l is large enough compared to k and k>6. The case when k<7 was already settled asymptotically by Kelly, Kuehn and Osthus.
Connectivity and Diameter of Friends and Strangers Graphs Using Minimum Degree
针对友敌图的连通性与直径问题，基于最小度条件，证明当图满足一定最小度要求时友敌图连通且直径达到最优界，改进了相关研究成果并解决了部分问题。
It was conjectured by Koh and Tay [Graphs Combin. 18(4) (2002), 745--756] that for $n\geq 5$ every simple graph of order $n$ and size at least $\binom{n}{2}-n+5$ has an orientation of diameter two. We prove this conjecture and hence determine for every $n\geq 5$ the minimum value of $m$ such that every graph of order $n$ and size $m$ has an orientation of diameter two.
A special case of a conjecture by Thomassé is that any oriented graph with minimum outdegree k contains a dipath of length 2k. For the sake of proving whether or not a counterexample exists, we present reductions and establish bounds on both connectivity and the number of vertices in a counterexample. We further conjecture that in any oriented graph with minimum outdegree k, every vertex belongs to a dipath of length 2k.
Oriented Diameter of Mixed Graphs with Given Maximum Undirected Degree
本文将无桥无向图及二部图的有向直径尖锐界推广到混合图，利用最大无向度等参数给出有向直径的紧上界，并证明除一个情形外所有界均为尖锐。
A sum graph is a finite simple graph whose vertex set is labeled with distinct positive integers such that two vertices are adjacent if and only if the sum of their labels is itself another label. The spum of a graph $G$ is the minimum difference between the largest and smallest labels in a sum graph consisting of $G$ and the minimum number of additional isolated vertices necessary so that a sum graph labeling exists. We investigate the spum of various families of graphs, namely cycles, paths, and matchings. We introduce the sum-diameter, a modification of the definition of spum that omits the…
We show that for each \ell\geq 4 every sufficiently large oriented graph G with δ^+(G), δ^-(G) \geq \lfloor |G|/3 \rfloor +1 contains an \ell-cycle. This is best possible for all those \ell\geq 4 which are not divisible by 3. Surprisingly, for some other values of \ell, an \ell-cycle is forced by a much weaker minimum degree condition. We propose and discuss a conjecture regarding the precise minimum degree which forces an \ell-cycle (with \ell \geq 4 divisible by 3) in an oriented graph. We also give an application of our results to pancyclicity and consider \ell-cycles in general digraphs.
Counting oriented trees in digraphs with large minimum semidegree
本文研究在有向图 $G$ 中包含多少个有向树 $T$ 的副本，证明了在最小半度条件下，$G$ 至少包含最优数量的 $T$ 的副本，并推广了无向图的相关结论。
Oriented Discrepancy of The Square of Hamilton Cycles
本文研究了在给定最小度数条件下，Hamilton回路的平方是否存在具有大正向与反向边不平衡的子图，证明了当n足够大时，每个具有最小度数δ(G)≥2n/3的定向图G都包含一个Hamilton回路的平方，其最大σ值超过依赖于δ(G)和n的函数。
Let $L(n,d)$ denote the minimum possible number of leaves in a tree of order $n$ and diameter $d.$ In 1975 Lesniak gave the lower bound $B(n,d)=\lceil 2(n-1)/d\rceil$ for $L(n,d).$ When $d$ is even, $B(n,d)=L(n,d).$ But when $d$ is odd, $B(n,d)$ is smaller than $L(n,d)$ in general. For example, $B(21,3)=14$ while $L(21,3)=19.$ We prove that for $d\ge 2,$ $ L(n,d)=\left\lceil \frac{2(n-1)}{d}\right\rceil$ if $d$ is even and $L(n,d)=\left\lceil \frac{2(n-2)}{d-1}\right\rceil$ if $d$ is odd. The converse problem is also considered. Let $D(n,f)$ be the minimum possible diameter of a tree of order …
Assignment of one of the two possible directions to every edge of an undirected graph $G=(V,E)$ is called an orientation of $G$. The resulting directed graph is denoted by $\overrightarrow{G}$. A strong orientation is one in which every vertex is reachable from every other vertex via a directed path. The diameter of $\overrightarrow{G}$, i.e., the maximum distance from one vertex to another, depends on the particular orientation. The minimum diameter among all possible orientations is called the oriented diameter $\overrightarrow{\text{diam}}(G)$ of $G$. Let $n,k$ be two integers with $1 \leq …
We use a randomised embedding method to prove that for all α>0 any sufficiently large oriented graph G with minimum in-degree and out-degree δ^+(G),δ^-(G)\geq (3/8+α)|G| contains every possible orientation of a Hamilton cycle. This confirms a conjecture of Häggkvist and Thomason.
Oriented Minimum spanning tree looks like the Uniform spanning tree on the complete graph (at least locally)
研究完全图中有向最小生成树形图的局部极限，证明其与均匀随机树局部极限相同，与最小生成树不同，揭示引入方向改变最小生成树局部几何结构。
In the complete graph on n vertices, when each edge has a weight which is an exponential random variable, Frieze proved that the minimum spanning tree has weight tending to zeta(3)=1/1^3+1/2^3+1/3^3+... as n goes to infinity. We consider spanning trees constrained to have depth bounded by k from a specified root. We prove that if k > log_2 log n+omega(1), where omega(1) is any function going to infinity with n, then the minimum bounded-depth spanning tree still has weight tending to zeta(3) as n -> infinity, and that if k < log_2 log n, then the weight is doubly-exponentially large in log_2 lo…
Oriented trees in digraphs without short non-directed cycles
本文将 Jiang 关于大围长图包含给定树的经典结果推广到有向图，在最小半度满足 $δ^0(D)\ge \max\{k/\ell,Δ(T)\}$ 的条件下证明大围长有向图包含所有 $k$ 边有向树，肯定回答了 Stein 和 Trujillo-Negrete 的问题。
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