A simple 7/3-approximation algorithm for feedback vertex set in tournaments

Vertex-disjoint cycles in tournaments

The Bermond-Thomassen conjecture states that, for any positive integer $r$, a digraph of minimum out-degree at least $2r-1$ contains at least $r$ vertex-disjoint directed cycles. Bessy, Sereni and Lichiardopol proved that a regular tournament $T$ of minimum degree $2r-1$ contains at least $r$ vertex-disjoint directed cycles, which shows that the above conjecture is true for tournaments. After that, Lichiardopol improved this result by showing that a $2r-1$-regular tournament contains at least $\frac{7}{6}r-\frac{7}{3}$ vertex-disjoint directed cycles. In this paper, we will extend the result t…

Vertex-disjoint directed cycles of prescribed length in tournaments with given minimum out-degree

The Bermond-Thomassen conjecture states that, for any positive integer $r$, a digraph of minimum out-degree at least $2r-1$ contains at least $r$ vertex-disjoint directed cycles. In 2014, Bang-Jensen, Bessy and Thomass\' e proved the conjecture for tournaments. In 2010, Lichiardopol conjectured that a tournament $T$ with minimum out-degree at least $(q-1)r-1$ contains at least $r$ vertex-disjoint $q$-cycles, where integer $q\geq3$ and $r\geq1$. In this paper, we address Lichiardopol's conjecture affirmatively. In particular, the case $q=3$ implies Bermond-Thomassen conjecture for tournaments.

Tournaments and the Strong Erdős-Hajnal Property

A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erdős-Hajnal Conjecture, states that for every tournament $S$ there exists $ε(S)>0$ such that if $T$ is an $n$-vertex tournament that does not contains $S$ as a subtournament, then $T$ contains a transitive subtournament on at least $n^{ε(S)}$ vertices. Let $C_5$ be the unique five-vertex tournament where every vertex has two inneighbors and two outneighbors. The Alon-Pach-Solymosi conjecture is known to be true for the case when $S=C_5$. Here we prove a strengthening of this result, showing that in every tournament…

The second out-neighbourhood for local tournaments

Sullivan stated the conjectures: (1) every oriented graph $D$ has a vertex $x$ such that $d^{++}(x)\geq d^{-}(x)$; (2) every oriented graph $D$ has a vertex $x$ such that $d^{++}(x)+d^{+}(x)\geq 2d^{-}(x)$. In this paper, we prove that these conjectures hold for local tournaments. In particular, for a local tournament $D$, we prove that $D$ has at least two vertices satisfying $(1)$ if $D$ has no vertex of in-degree zero. And, for a local tournament $D$, we prove that either there exist two vertices satisfying $(2)$ or there exists a vertex $v$ satisfying $d^{++}(v)+d^{+}(v)\geq 2d^{-}(v)+2$ i…

关于三色锦标赛

On 3-coloured tournaments

我们（重新）证明，在每一个3-边着色的比赛中，没有顶点是事件与所有的颜色有一个循环的彩虹三角形或一个顶点支配其他顶点单色。

Counting orientations of graphs with no strongly connected tournaments

Let $S_k(n)$ be the maximum number of orientations of an $n$-vertex graph $G$ in which no copy of $K_k$ is strongly connected. For all integers $n$, $k\geq 4$ where $n\geq 5$ or $k\geq 5$, we prove that $S_k(n) = 2^{t_{k-1}(n)}$, where $t_{k-1}(n)$ is the number of edges of the $n$-vertex $(k-1)$-partite Turán graph $T_{k-1}(n)$, and that $T_{k-1}(n)$ is the only $n$-vertex graph with this number of orientations. Furthermore, $S_4(4) = 40$ and this maximality is achieved only by $K_4$.

7-泛锦标赛的新上界

A New Upper Bound for 7-Universal Tournaments

本文构造了一个14顶点的显式锦标赛，包含所有7顶点锦标赛作为诱导子锦标赛，通过穷举验证，将上界从15改进至14，从而确定最小阶数为13或14。

Trees with few leaves in tournaments

We prove that there exists $C>0$ such that any $(n+Ck)$-vertex tournament contains a copy of every $n$-vertex oriented tree with $k$ leaves, improving the previously best known bound of $n+O(k^2)$ vertices to give a result tight up to the value of $C$. Furthermore, we show that, for each $k$, there exists $n_0$, such that, whenever $n\geqslant n_0$, any $(n+k-2)$-vertex tournament contains a copy of every $n$-vertex oriented tree with at most $k$ leaves, confirming a conjecture of Dross and Havet.

Common kings of a chain of cycles in a strong tournament

It is known that every strong tournament has directed cycles of any length, and thereby strong subtournaments of any size. In this note, we prove that they also can share a common vertex which is a king of all of them. This common vertex can be any king in the whole tournament. Further, the Hamiltonian cycles in them can be recursively constructed by inserting an additional vertex to one directed edge.

Locating Dominating Sets in local tournaments

A dominating set in a directed graph is a set of vertices $S$ such that all the vertices that do not belong to $S$ have an in-neighbour in $S$. A locating set $S$ is a set of vertices such that all the vertices that do not belong to $S$ are characterized uniquely by the in-neighbours they have in $S$, i.e. for every two vertices $u$ and $v$ that are not in $S$, there exists a vertex $s\in S$ that dominates exactly one of them. The size of a smallest set of a directed graph $D$ which is both locating and dominating is denoted by $γ^{LD}(D)$. Foucaud, Heydarshahi and Parreau proved that any twin…

关于Tourist的几个定理

Various Theorems on Tournaments

在这篇论文中，我们证明了竞赛图的各种定理。\emph{prime}锦标赛是这样一种锦标赛$G$，即没有$X \subseteq V(G)$、$1 < |X| < |V(G)|$，即对于每个顶点$v \in V(G) \minus X$，要么是所有$x \in X$的$v \ra x$，要么是所有$x \in X$的$x \ra v$。首先，我们证明了给定一个不属于三个特殊竞赛图族之一的素竞赛图$G$，对于$G$的任何一个素子竞赛图$H$，若$G$具有$5 \le |V(H)| < |V(G)|$，则存在$G$的一个素子竞赛图，其顶点具有与$H$同构的$|V(H)| + 1$子竞赛图.我们接下来证明，对于素数竞赛图$G$中的任何两个循环三角形$C$，$C^\prime$，存在循环三角形序列$C_1,...,C_n$，使得$C_1 = C$，$C_n = C^\prime$和$C_i$与$C_{i+1}$共享一条边，对于所有$1 \le i \le n-1$。接下来，我们考虑我们称之为\emph{matching tournaments}的竞赛图，它的顶点可以在一条水平线上排序，这样每个顶点最多是一条从右向左的边的头或尾。我们确定了锦标赛可以有满足上述条件的两种不同顺序的条件。我们还证明了，有无穷多个最小的比赛，不匹配的比赛。最后，我们考虑的比赛$K_n$和$K_n^\ast$，这是从…

When you come at the kings you best not miss

A tournament is an orientation of a complete graph. We say that a vertex $x$ in a tournament $\vec T$ controls another vertex $y$ if there exists a directed path of length at most two from $x$ to $y$. A vertex is called a king if it controls every vertex of the tournament. It is well known that every tournament has a king. We follow Shen, Sheng, and Wu (SIAM J. Comput., 2003) in investigating the query complexity of finding a king, that is, the number of arcs in $\vec T$ one has to know in order to surely identify at least one vertex as a king. The aforementioned authors showed that one always…

竞赛图中边不交传递三元组的个数

The number of edge disjoint transitive triples in a tournament

我们证明了一个具有$n$顶点的竞赛图有多于$0.13n^2(1+o(1))$的边不交传递三元组。证明了在$n$ -顶点竞赛图中$k$ -顶点传递竞赛图的大填充的存在性。我们的证明结合概率参数和一些强大的包装由于威尔逊和弗兰克尔和Rödl的结果。

关于正则锦标赛中缺失的近生成平衡反定向树的注记

Note on near-spanning balanced antidirected trees missing from regular tournaments

本文构造了满足半度条件但无法嵌入高度对称锦标赛的近生成平衡反定向树，给出了Stein猜想在均匀次线性度解释下的反例，并确定了最优目标阶及最小宿主阶。

On 1-subdivisions of transitive tournaments

The oriented Ramsey number $\vec{r}(H)$ for an acyclic digraph $H$ is the minimum integer $n$ such that any $n$-vertex tournament contains a copy of $H$ as a subgraph. We prove that the $1$-subdivision of the $k$-vertex transitive tournament $H_k$ satisfies $\vec{r}(H_k)= O(k^2\log\log k)$. This is tight up to multiplicative $\log\log k$-term. We also show that if $T$ is an $n$-vertex tournament with $Δ^+(T)-δ^+(T)= O(n/k) - k^2$, then $T$ contains a $1$-subdivision of $\vec{K}_k$, a complete $k$-vertex digraph with all possible $k(k-1)$ arcs. This is also tight up to multiplicative constant.

具有一条仅位于唯一有向三角形中的弧的 Walecki 竞赛图

Walecki tournaments with an arc that lies in a unique directed triangle

本文研究含唯一有向三角形弧的 Walecki 竞赛图，证明标记为 * 的顶点被所有自同构固定且同构间相互对应，并给出至少 2^k 个适用签名。

3-锦标赛中的统治

Domination in 3-tournaments

一个3-竞赛图是一个完全的3-一致超图，其中每条边都有一个特殊的顶点作为它的尾。顶点集$X$支配$T$，如果不在$X$中的每个顶点都包含在其尾部在$X$中的边中。$T$的支配数是这样的$X$的最小尺寸。Gyárfás推广了关于通常（图）竞赛图的著名结果，证明了存在具有任意大控制数的3-竞赛图，并且如果任意四个顶点诱导两个具有相同尾的三元组，则情况并非如此。在这个简短的说明中，我们解决了这两个问题，证明了第一个猜想，并反驳了第二个。

竞赛图中长度不同的顶点不交圈

Vertex-disjoint cycles of different lengths in tournaments

本研究围绕竞赛图中不同长度顶点不交圈的猜想展开，证明当k≥5时最小出度≥2k-1的竞赛图含k个顶点不交圈且其中3个长度不同，还证实最小出度≥6的竞赛图含3个不同长度的顶点不交圈且条件紧，解答了Chen与Chang的问题。

Forests and the Strong Erdos-Hajnal Property

An equivalent directed version of the celebrated unresolved conjecture of Erdos and Hajnal proposed by Alon et al. states that for every tournament H there exists epsilon(H) > 0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least n^(epsilon(H)). A tournament H has the strong EH-property if there exists c > 0 such that for every H-free tournament T with |T| > 1, there exist disjoint vertex subsets A and B, each of cardinality at least c|T| and every vertex of A is adjacent to every vertex of B. Berger et al. proved that the unique five-vertex tourn…

用小标签类区分旅游者

Distinguishing Tournaments with Small Label Classes

有向图的$d$ -可区别点（弧）标号是使用不被任何非平凡自同构保持的$d$标号的点（弧）标号。令$ρ(T)$（$ρ'(T)$）是锦标赛$T$的2-可区别顶点（弧）标记中的标记类的最小大小。Gluck定理意味着对于任何$n$阶的竞赛图$T$，$ρ(T) \le \lfloor n/2 \rfloor$。本文构造了一类竞赛图$\cal H$，使得对于$\cal H$中的任意阶$n$竞赛图，$ρ(T) \ge \lfloor n/2 \rfloor$.另外，我们证明了当$T \in {\cal H}$和有$n$阶时，对于任意$T$的$n$和$ρ'(T) \ge \lceil n/6 \rceil$的竞赛图，$ρ'(T) \le \lfloor 7n/36 \rfloor + 3$都是.这些结果回答了Boutin提出的一些悬而未决的问题。

Dominant tournament families

For a tournament $H$ with $h$ vertices, its typical density is $h!2^{-\binom{h}{2}}/aut(H)$, i.e. this is the expected density of $H$ in a random tournament. A family ${\mathcal F}$ of $h$-vertex tournaments is {\em dominant} if for all sufficiently large $n$, there exists an $n$-vertex tournament $G$ such that the density of each element of ${\mathcal F}$ in $G$ is larger than its typical density by a constant factor. Characterizing all dominant families is challenging already for small $h$. Here we characterize several large dominant families for every $h$. In particular, we prove the follow…

Sparse spanning $k$-connected subgraphs in tournaments

In 2009, Bang-Jensen asked whether there exists a function $g(k)$ such that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + g(k)$ arcs. In this paper, we answer the question by showing that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + 750k^2\log(k+1)$ arcs.

Kernels by rainbow paths in arc-colored tournaments

For an arc-colored digraph $D$, define its {\em kernel by rainbow paths} to be a set $S$ of vertices such that (i) no two vertices of $S$ are connected by a rainbow path in $D$, and (ii) every vertex outside $S$ can reach $S$ by a rainbow path in $D$. In this paper, we show that it is NP-complete to decide whether an arc-colored tournament has a kernel by rainbow paths, where a {\em tournament} is an orientation of a complete graph. In addition, we show that every arc-colored $n$-vertex tournament with all its strongly connected $k$-vertex subtournaments, $3\leq k\leq n$, colored with at least…

Cyclic triangle factors in regular tournaments

Both Cuckler and Yuster independently conjectured that when $n$ is an odd positive multiple of $3$ every regular tournament on $n$ vertices contains a collection of $n/3$ vertex-disjoint copies of the cyclic triangle. Soon after, Keevash and Sudakov proved that if $G$ is an orientation of a graph on $n$ vertices in which every vertex has both indegree and outdegree at least $(1/2 - o(1))n$, then there exists a collection of vertex-disjoint cyclic triangles that covers all but at most $3$ vertices. In this paper, we resolve the conjecture of Cuckler and Yuster for sufficiently large $n$.

Galaxies and the Strong Erdos-Hajnal Property

An equivalent directed version of the celebrated unresolved conjecture of Erdos and Hajnal proposed by Alon, Pack, and Solymosi states that for every tournament H there exists epsilon(H)>0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least n^(epsilon(H)). A tournament H has the strong EH-property if there exists c > 0 such that for every H-free tournament T with |T| > 1, there exist disjoint vertex subsets A and B, each of cardinality at least |T|n and every vertex of A is adjacent to every vertex of B. Berger et al. proved that the unique five-v…

关于高维无圈竞赛图

On high-dimensional acyclic tournaments

我们研究了一个高维模拟的概念，一个非循环（又名传递）比赛。给出了$d$ -维$n$ -点无圈竞赛图个数的上下界.此外，我们证明了每个$n$ -顶点$d$ -维竞赛图包含一个$Ω(\log^{1/d}n)$顶点的非循环子竞赛图，并且有界是紧的。此声明用于锦标赛（即，情况$d=1$）是众所周知的事实。我们指出了超图的无圈高维竞赛图和Ramsey数之间的联系。我们调查以及其他各种概念之间的相互关系，在高维比赛的无环性。这些包括组合，几何和拓扑概念。

竞赛图中具有许多叶子的树

Trees with many leaves in tournaments

本文研究竞赛图中包含有向树的条件，基于叶子数量和最大度数改进了Sumner通用竞赛图猜想，证明了Havet和Thomassé猜想的渐近形式，并改进了Mycroft和Naia关于最大度数限制的结果。

竞赛图的星因子

Star-factors of tournaments

设S_m表示m-1条有公共尾边构成的m-顶点简单有向图。设f（m）表示最小n，使得每个n-顶点竞赛图都有一个生成子图，生成子图由S_m的n/m个不相交副本组成.我们证明了对于充分大的m，mlgm-mlglgm <= f（m）<= 4m^2 - 6m.

tournaments 的 Schuttes 性质及在骰子游戏中的应用

Schuttes property for sets of tournaments and an application to dice games

研究了 tournaments 的 Schuttes 性质的推广，探讨了 m 个 tournaments 需要的最少顶点数，并应用于设计新的骰子游戏。

Chickens and Dukes

Following on the King Chicken Theorems originally proved by Maurer, we examine the idea of multiple flocks of chickens by bringing the chickens from tournaments to multipartite tournaments. As Kings have already been studied in multipartite settings, notably by Koh-Tan and Petrovic-Thomassen, we examine a new type of chicken more suited than Kings for these multipartite graphs: Dukes. We define an M-Duke to be a vertex from which any vertex in a different partite set is accessible by a directed path of length at most M. In analogy with Maurer's paper, we prove various structural results regard…

通过弧邻域界定稠密有向图的色数

Bounding the chromatic number of dense digraphs by arc neighborhoods

本文证明竞赛图中所有弧邻域色数有界则整图色数有界，并推广到独立数有界的稠密定向图，由此建立两个高色数结构猜想的等价性。

随机旅游图和有向图的Seymour点数

The Number of Seymour Vertices in Random Tournaments and Digraphs

西摩距离2猜想指出，在任何有向图中，都存在一个顶点（“西摩顶点”），它在距离2处的邻居至少和在距离1处的邻居一样多。我们探索的有效性的概率语句沿线建议西摩的猜想，证明几乎肯定有一个“大”的西摩顶点的随机比赛和“甚至更多”一般随机有向图。

k边着色竞赛图中单色路径的国王农奴二人组

King-serf duo by monochromatic paths in k-edge-coloured tournaments

一个开放的猜想Erdans的状态，对于每一个正整数$k$有一个（最小）正整数$f(k)$，使每当一个锦标赛有它的边着色$k$颜色，有一个集合$S$最多$f(k)$顶点，使每个顶点有一个单色路径到某些点在$S$。我们考虑了一个相关的问题，并表明，对于每个（有限或无限）基数$κ>0$有一个基数$ λ_κ$，使在每个$κ$ -边色锦标赛存在不相交的顶点集$K,S$的总大小最多为$ λ_κ$，使每个顶点$ v $有一个单色路径的长度最多为2从$K$到$v$或从$v$到$S$。

$Δ(1, 2, 2)$-free 竞赛图的结构

The structure of $Δ(1, 2, 2)$-free tournaments

本文扩展了排除特定子竞赛图的竞赛图结构描述，证明了排除 $Δ(1, 2, 2)$ 的竞赛图由三种小竞赛图或两种操作生成，并给出了色数、最大可迁子竞赛图大小和循环三角形数量的最优界限。

弧色竞赛图中吸收集大小的一个下界

A lower bound on the size of an absorbing set in an arc-coloured tournament

Bousquet，Lochet和Mr.Bassé最近给出了一个优雅的证明，对于任何整数$n$，有一个最小整数$f(n)$，使任何锦标赛的弧着色与$n$颜色包含一个子集的顶点$S$的大小为$f(n)$的性质，任何顶点不在$S$承认一个单色路径的一些顶点的$S$。在本说明中，我们提供了值$f(n)$的下界。

可行锦标赛画像区域的维数

The dimension of the region of feasible tournament profiles

本文确定了 k 顶点锦标赛画像区域的维数，发现其远大于强连通锦标赛数量，并揭示了该问题与 Lyndon 词之间的有趣联系。

Transitive tournament tilings in oriented graphs with large minimum total degree

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.

Paths of given length in tournaments

We prove that every $n$-vertex tournament has at most $n\left(\frac{n-1}{2}\right)^k$ walks of length $k$.

Rainbow triangles in arc-colored tournaments

Let $T_{n}$ be an arc-colored tournament of order $n$. The maximum monochromatic indegree $Δ^{-mon}(T_{n})$ (resp. outdegree $Δ^{+mon}(T_{n})$) of $T_{n}$ is the maximum number of in-arcs (resp. out-arcs) of a same color incident to a vertex of $T_{n}$. The irregularity $i(T_{n})$ of $T_{n}$ is the maximum difference between the indegree and outdegree of a vertex of $T_{n}$. A subdigraph $H$ of an arc-colored digraph $D$ is called rainbow if each pair of arcs in $H$ have distinct colors. In this paper, we show that each vertex $v$ in an arc-colored tournament $T_{n}$ with $Δ^{-mon}(T_n)\leqΔ^{…

竞赛图划分猜想的证明及其对具有指定圈长的1-因子的应用

Proof of a tournament partition conjecture and an application to 1-factors with prescribed cycle lengths

在1982年，Schlassen提出了一个问题：是否存在一个整数f（k，t）使得每个强f（k，t）-连通竞赛图T允许它的顶点集被划分为t个顶点类V_1，...，V_t使得对所有i，V_i在T上诱导的子竞赛图T[V_i]是强k-连通的。我们的主要结果意味着对这个问题的肯定回答。特别地，我们证明了f（k，t）= O（k^7t^4）就足够了。作为我们的主要结果的另一个应用，我们给一个肯定的回答宋的一个问题，是否，对于任何整数t，存在一个整数h（t），使每个强h（t）-连通竞赛有一个1-因子组成的t顶点不相交的周期的规定长度。我们证明了h（t）= O（t^5）就足够了。

具有竞赛图的平铺有向图

Tiling directed graphs with tournaments

Hajnal-Szemerédi定理指出，对于任何整数$r \ge 1$和$r$的任何倍数$n$，如果$G$是在$n$顶点和$δ(G) \ge (1 - 1/r)n$上的图，则$G$可以被划分为在$r$顶点上的完全图的$n/r$顶点不相交的副本。对于有向图，我们证明了一个非常一般的类似结果：对于任何整数$r \ge 4$和$r$的任何足够大的多个$n$，如果$G$是$n$顶点上的有向图，并且每个顶点至少与$2(1 - 1/r)n - 1$有向边关联，则$G$可以被划分为大小为$r$的$n/r$顶点不相交子图，每个子图包含$r$顶点上的每个锦标赛。一个相关的Turán型结果也得到了证明。

着色比赛：从本地到全球

Coloring tournaments: from local to global

有向图$D$的\emph{chromatic number}是对$D$的顶点着色所需的最小颜色数，使得$D$的每个颜色类都导出一个无环子有向图。因此，竞赛图$T$的色数是覆盖$T$顶点集的传递子竞赛图的最小数目。在本文中，我们表明，比赛显着简单，比图着色。事实上，虽然无向图可以完全“局部简单”（每个邻域是一个独立集），并有大的色数，我们表明，局部简单的比赛确实是简单的。特别地，存在函数$f$，使得如果竞赛图$T$中的每个顶点的外邻域具有至多$c$的色数，则$T$具有至多$f(c)$的色数。这回答了Berger等人的一个问题。

All feedback arc sets of a random Turán tournament have n/k-k+1 disjoint k-cliques (and this is tight)

We look at structures that must be removed (or reversed) in order to make acyclic a given oriented graph. For a directed acyclic graph $H$ and an oriented graph $G$, let $f_H(G)$ be the maximum number of pairwise disjoint copies of $H$ that can be found in {\em all} feedback arc sets of $G$. In particular, to make $G$ acyclic, one must remove (or reverse) $f_H(G)$ pairwise disjoint copies of $H$. Most intriguing is the case where $H$ is a $k$-clique, where the parameter is denoted by $f_k(G)$. Determining $f_k(G)$ for arbitrary $G$ seems challenging. Here we determine $f_k(G)$ precisely for al…

锦标赛排序：对偶性与效率

Tournament Ranking: Duality and Efficiency

针对循环门杰（CM）锦标赛，提出多项式时间组合算法求解最小反馈弧集和最大循环打包问题，解决了此前仅知结构刻画而无算法的局限。

关于缺失两颗星或不相交路径的竞赛图的第二邻域猜想

About the second neighborhood conjecture for tournaments missing two stars or disjoint paths

本文针对 Seymour 第二邻域猜想，证明了缺失两颗星的竞赛图满足该猜想，并研究了缺失不相交路径（尤其是长度为 2 的路径）的竞赛图，在某些情形下还证明了至少存在两个具有第二邻域性质的顶点。

Spanning eulerian subdigraphs avoiding k prescribed arcs in tournaments

A digraph is {\bf eulerian} if it is connected and every vertex has its in-degree equal to its out-degree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. A digraph is {\bf semicomplete} if it has no pair of non-adjacent vertices. A {\bf tournament} is a semicomplete digraph without directed cycles of length 2. Fraise and Thomassen \cite{fraisseGC3} proved that every $(k+1)$-strong tournament has a hamiltonian cycle which avoids any prescribed set of $k$ arcs. In \cite{bangsupereuler} the authors demonstrated that a number of results concerning vertex-c…

高色数竞赛图的无环子图

Acyclic subgraphs of tournaments with high chromatic number

本文证明了每个 $n$ 顶点竞赛图存在色数至少为 $n^{5/9-o(1)}$ 的无环子图，且上界为 $n^{3/4+o(1)}$，强有力地证实并改进了 Nassar 和 Yuster 的猜想，其证明结合了概率与谱技术。

纯对。X. 锦标赛与强 Erdős-Hajnal 性质

Pure pairs. X. Tournaments and the strong Erdos-Hajnal property

本文研究锦标赛中纯对与强 Erdős-Hajnal 性质的关系，证明了至多三条回边的锦标赛具有强 EH 性质，并给出一个七顶点反例，从而扩展了关于至多六顶点锦标赛的已有结果。

路径可扩展竞赛图

Path Extendable Tournaments

本文研究竞赛图的路径可扩展性，证明了基于不规则度 $i(T)$ 和长度为2路径最小数量 $\pi_2(T)$ 的两个充分条件，并推导出几乎所有随机竞赛图均路径可扩展。

关于Tourist的移去引理

The Removal Lemma for Tournaments

假设需要改变$n$ -顶点锦标赛$T$的至少$εn^2$边的方向，以便使其无$H$。正则性方法的标准应用表明，在这种情况下，$T$至少包含$H$的$f^*_H(ε)n^h$拷贝，其中$f^*_H$是某种塔型函数。长期以来，人们已经观察到，许多图/有向图的问题变得更容易时，假设主机图是一个锦标赛。因此很自然地会问，如果我们假设有向图$G$是一个竞赛图，移除引理是否变得更容易。我们的主要结果在这里是一个精确的表征的比赛$H$的$f^*_H(ε)$是多项式在$ε$，说明这样的界是可达到的，当且仅当$H$的顶点集可以划分为两个集，每个跨越一个无环有向图。这个特征的证明依赖于阿隆、费舍尔和纽曼的矩阵正则性引理的新应用，以及鲁萨-塞梅雷迪图的概率变体。我们最后表明，即使限制比赛，如果$H$满足我们的特征条件的决定是一个NP-困难的问题。

平铺传递竞赛图及其blow-up

Tiling transitive tournaments and their blow-ups

令$TT_k$表示$k$顶点上的传递竞赛图。令$TT(h,k)$表示通过用大小为$h \geq 1$的独立集合替换每个顶点而从$TT_k$获得的图。证明了以下结果：设$c_2=1/2$，$c_3=5/6$和$c_k=1-2^{-k-\log k}$为$k \geq 4$ .对于每个$ε> 0$，存在$N=N(ε,h,k)$，使得对于每个具有$n > N$顶点和$δ(G) \geq c_kn$的无向图$G$，$G$的每个方向都包含$TT(h,k)$的顶点不相交副本，这些副本覆盖除最多$εn$顶点之外的所有顶点。在$k=2$和$k=3$的情况下，结果是渐近紧的。对于$k \geq 4$，$c_k$不能改善至小于$1-2^{-0.5k(1+o(1))}$。

寻找国王与强国王的难度

Hardness of Finding Kings and Strong Kings

本文研究了有向图中寻找国王与强国王的查询复杂度，证明了在任意 $n$ 顶点有向图中确定国王存在性及在竞赛图中寻找强国王的查询复杂度均为 $Θ(n^2)$，并设计了特定竞赛图以辅助证明。

色数并非锦标赛局部可判定

Chromatic number is not tournament-local

本文否定 Scott 与 Seymour 关于色数可由锦标赛局部邻域刻画的猜想，并进一步否定相关推广问题，同时揭示退化性情形具有不同行为。

Arc reversals of cycles in orientations of $G$ vertex-multiplications

Ryser proved that any two tournaments with the same score sequence are $C_3$-equivalent while Beineke and Moon proved the $C_4$-equivalence for any two bipartite tournaments with the same score lists. In this paper, we extend these results to orientations of $G$ vertex-multiplications. We focus on two main areas, namely orientations with the same score list and with score-list parity. Our main tools are extensions of the refinement technique, directed difference graph and a reduction lemma.