Raphael Yuster
We prove that a tournament with $n$ vertices has more than $0.13n^2(1+o(1))$ edge-disjoint transitive triples. We also prove some results on the existence of large packings of $k$-vertex transitive tournaments in an $n$-vertex tournament. Our proofs combine probabilistic arguments and some powerful packing results due to Wilson and to Frankl and Rödl.
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We construct a combinatorial function F which computes the number of oriented Hamiltonian paths of any given type, in a transitive tournament. We also study many properties of F that arise, and reach some observations.
We prove the existence of a function $h(k)$ such that every simple digraph with minimum outdegree greater than $h(k)$ contains an immersion of the transitive tournament on $k$ vertices. This solves a conjecture of Devos, McDonald, Mohar and Scheide.
We prove that for every fixed $k$, the number of occurrences of the transitive tournament $Tr_k$ of order $k$ in a tournament $T_n$ on $n$ vertices is asymptotically minimized when $T_n$ is random. In the opposite direction, we show that any sequence of tournaments $\{T_n\}$ achieving this minimum for any fixed $k\geq 4$ is necessarily quasi-random. We present several other characterizations of quasi-random tournaments nicely complementing previously known results and relatively easily following from our proof techniques.
We further study sets of labeled dice in which the relation "is a better die than" is non-transitive. Focusing on sets with an additional symmetry we call "balance," we prove that sets of $n$ such $m$-sided dice exist for all $n,m \geq 3$. We then show how to construct a set of $n$ dice such that the relation behaves according to the direction of the arrows of any tournament (complete directed graph) on $n$ vertices.
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.
Chain--collider--fork Decompositions of Transitive Tournament
研究传递锦标赛中链、碰撞和叉三种两弧基序的分解与打包，给出最大弧不交拷贝数。
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.
A tournament is called locally transitive if the outneighbourhood and the inneighbourhood of every vertex are transitive. Equivalently, a tournament is locally transitive if it avoids the tournaments $W_4$ and $L_4$, which are the only tournaments up to isomorphism on four vertices containing a unique $3$-cycle. On the other hand, a sequence of tournaments $(T_n)_{n\in\mathbb{N}}$ with $|V(T_n)| = n$ is called almost balanced if all but $o(n)$ vertices of $T_n$ have outdegree $(1/2 + o(1))n$. In the same spirit of quasi-random properties, we present several characterizations of tournament sequ…
In an earlier paper the first two authors have shown that self-complementary graphs can always be oriented in such a way that the union of the oriented version and its isomorphically oriented complement gives a transitive tournament. We investigate the possibilities of generalizing this theorem to decompositions of the complete graph into three or more isomorphic graphs. We find that a complete characterization of when an orientation with similar properties is possible seems elusive. Nevertheless, we give sufficient conditions that generalize the earlier theorem and also imply that decompositi…
The study of problems concerning subdivisions of graphs has a rich history in extremal combinatorics. Confirming a conjecture of Burr and Erdős, Alon proved in 1994 that subdivided graphs have linear Ramsey numbers. Later, Alon, Krivelevich and Sudakov showed that every $n$-vertex graph with at least $\varepsilon n^2$ edges contains a $1$-subdivision of the complete graph on $c_{\varepsilon}\sqrt{n}$ vertices, resolving another old conjecture of Erdős. In this paper we consider the directed analogue of these problems and show that every tournament on at least $(2+o(1))k^2$ vertices contains th…
Ladder tournaments are widely used to rank individuals in real-world organizations and games. Their mathematical properties however are still poorly understood. We formalize the ranking rule generated by a ladder tournament, and we show that it is neither complete nor transitive in general. If it is complete, then it is transitive and its asymmetric component is a finite union of transitive tournaments. We also study the relationship between an individual's rank and his performance as measured by the frequency at which he is pivotal. We show an individual's pivotability is a weakly increasing …
Tournaments are graphs obtained by assigning a direction for every edge in an undirected complete graph. We give a formula for the number of isomorphism classes of vertex-transitive tournaments with prime order. For that, we introduce Cayley tournaments, which are special Cayley digraphs, and show that there is a one to one identification between the isomorphism classes of vertex-transitive tournaments with prime order and the isomorphism classes of Cayley tournaments with prime order.
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…
In 2010, N. Lichiardopol conjectured for $q \geq 3$ and $k \geq 1$ that any tournament with minimum out-degree at least $(q-1)k-1$ contains $k$ disjoint cycles of length $q$. We prove this conjecture for $q \geq 5$. Since it is already known to hold for $q\le4$, this completes the proof of the conjecture.
Let $TT_k$ denote the transitive tournament on $k$ vertices. Let $TT(h,k)$ denote the graph obtained from $TT_k$ by replacing each vertex with an independent set of size $h \geq 1$. The following result is proved: Let $c_2=1/2$, $c_3=5/6$ and $c_k=1-2^{-k-\log k}$ for $k \geq 4$. For every $ε> 0$ there exists $N=N(ε,h,k)$ such that for every undirected graph $G$ with $n > N$ vertices and with $δ(G) \geq c_kn$, every orientation of $G$ contains vertex disjoint copies of $TT(h,k)$ that cover all but at most $εn$ vertices. In the cases $k=2$ and $k=3$ the result is asymptotically tight. For $k \g…
On Ramsey Properties of k-Majority Tournaments
本文研究了k-多数竞赛的拉姆齐性质，证明了此类竞赛包含较大的传递子图，并提出了关于随机k-多数竞赛的开放问题。
If $T$ is an $n$-vertex tournament with a given number of $3$-cycles, what can be said about the number of its $4$-cycles? The most interesting range of this problem is where $T$ is assumed to have $c\cdot n^3$ cyclic triples for some $c>0$ and we seek to minimize the number of $4$-cycles. We conjecture that the (asymptotic) minimizing $T$ is a random blow-up of a constant-sized transitive tournament. Using the method of flag algebras, we derive a lower bound that almost matches the conjectured value. We are able to answer the easier problem of maximizing the number of $4$-cycles. These questi…
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.
A tournament is $k$-spectrally monomorphic if all the $k\times k$ principal submatrices of its adjacency matrix have the same characteristic polynomial. Transitive $n$-tournaments are trivially $k$-spectrally monomorphic. We show that there are no other for $k\in \{3,\ldots,n-3\} $. Furthermore, we prove that for $n\geq 5$, a non-transitive $n$-tournament is $(n-2)$-spectrally monomorphic if and only if it is doubly regular. Finally, we give some results on $(n-1)$-spectrally monomorphic regular tournaments.
Thomassen conjectured that there is a function $f(k)$ such that every strongly $f(k)$-connected tournament contains $k$ edge-disjoint Hamiltonian cycles. This conjecture was recently proved by Kühn, Lapinskas, Osthus, and Patel who showed that $f(k)\leq O(k^2(\log k)^2)$ and conjectured that there is a constant $C$ such that $f(k)\leq Ck^2$. We prove this conjecture.
We prove Seymour's Second Neighborhood Conjecture when the missing graph is disjoint stars under some conditions. Weaker conditions are required when n=2 or 3. In some cases, we exhibit two vertices with the desired property.
Note on Disjoint Cycles in Multipartite Tournaments
本文针对无三角形多部竞赛图和3部竞赛图验证了Bermond-Thomassen关于顶点不相交有向圈的猜想，并进一步刻画了最小出度至少为$2k-1$且任意$k$个不相交圈等长的3部竞赛图。
The typical structure of oriented graphs and digraphs with forbidden blow-up of transitive tournaments
研究禁止转置竞赛 tournaments 的吹胀体的有向图和有向图的典型结构，证明几乎所有此类图是 r 分支的，扩展了 Kuhn 等人的结果，验证了 Cherlin 的猜想。
Given $k$ pairs of vertices $(s_i,t_i)$, $1\le i\le k$, of a digraph $G$, how can we test whether there exist $k$ vertex-disjoint directed paths from $s_i$ to $t_i$ for $1\le i\le k$? This is NP-complete in general digraphs, even for $k = 2$, but for $k=2$ there is a polynomial-time algorithm when $G$ is a tournament (or more generally, a semicomplete digraph), due to Bang-Jensen and Thomassen. Here we prove that for all fixed $k$ there is a polynomial-time algorithm to solve the problem when $G$ is semicomplete.
The Erdős-Hajnal conjecture states that for every given undirected graph $H$ there exists a constant $c(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $|V(G)|^{c(H)}$. The conjecture is still open. Its equivalent directed version states that for every given tournament $H$ there exists a constant $c(H)>0$ such that every $H$-free tournament $T$ contains a transitive subtournament of order at least $|V(T)|^{c(H)}$. We prove in this paper that $\{H_{1},H_{2}\}$-free tournaments $T$ contain transitive subtournamen…
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。
Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments
本文通过禁止子有向图刻画了广义锦标赛和广义二分锦标赛若干族中的核完美与临界核不完美有向图，并证明判定CKI问题可多项式求解，回应了Bang-Jensen的猜想。
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。
Given $k$ pairs of vertices $(s_i,t_i)\;(1\le i\le k)$ of a digraph $G$, how can we test whether there exist vertex-disjoint directed paths from $s_i$ to $t_i$ for $1\le i\le k$? This is NP-complete in general digraphs, even for $k = 2$, but in an earlier paper we proved that for all fixed $k$, there is a polynomial-time algorithm to solve the problem if $G$ is a tournament (or more generally, a semicomplete digraph). Here we prove that for all fixed $k$ there is a polynomial-time algorithm to solve the problem when $V(G)$ is partitioned into a bounded number of sets each inducing a semicomple…
A tournament H is quasirandom-forcing if the following holds for every sequence (G_n) of tournaments of growing orders: if the density of H in G_n converges to the expected density of H in a random tournament, then (G_n) is quasirandom. Every transitive tournament with at least 4 vertices is quasirandom-forcing, and Coregliano et al. [Electron. J. Combin. 26 (2019), P1.44] showed that there is also a non-transitive 5-vertex tournament with the property. We show that no additional tournament has this property. This extends the result of Bucic et al. [Combinatorica 41 (2021), 175-208] that the n…
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…
We show that every sufficiently large regular tournament can almost completely be decomposed into edge-disjoint Hamilton cycles. More precisely, for each η>0 every regular tournament G of sufficiently large order n contains at least (1/2-η)n edge-disjoint Hamilton cycles. This gives an approximate solution to a conjecture of Kelly from 1968. Our result also extends to almost regular tournaments.
An Exact Dominant Degree Condition for Transitive Tournament Factors in Digraphs
该有向图理论研究给出了传递竞赛图因子的精确主导度条件，解决了Treglown的猜想与Molla等人的问题，得到了相关定理的更强版本。
An edge coloring of a tournament $T$ with colors $1,2,\dots,k$ is called \it $k$-transitive \rm if the digraph $T(i)$ defined by the edges of color $i$ is transitively oriented for each $1\le i \le k$. We explore a conjecture of the second author: For each positive integer $k$ there exists a (least) $p(k)$ such that every $k$-transitive tournament has a dominating set of at most $p(k)$ vertices. We show how this conjecture relates to other conjectures and results. For example, it is a special case of a well-known conjecture of Erd\H os, Sands, Sauer and Woodrow (so the conjecture is interestin…
Motivated by his work on the classification of countable homogeneous oriented graphs, Cherlin asked about the typical structure of oriented graphs (i) without a transitive triangle, or (ii) without an oriented triangle. We give an answer to these questions (which is not quite the predicted one). Our approach is based on the recent `hypergraph containers' method, developed independently by Saxton and Thomason as well as by Balogh, Morris and Samotij. Moreover, our results generalise to forbidden transitive tournaments and forbidden oriented cycles of any order, and also apply to digraphs. Along…
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。
Brandt et al. (2013) have recently disproved a conjecture by Schwartz (1990) by non-constructively showing the existence of a counterexample with about 10^136 alternatives. We provide a concrete counterexample for Schwartz's conjecture with only 24 alternatives.
We prove that every Eulerian orientation of $K_{m,n}$ contains $\frac{1}{4+\sqrt{8}}mn(1-o(1))$ arc-disjoint directed 4-cycles, improving earlier lower bounds. Combined with a probabilistic argument, this result is used to prove that every regular tournament with $n$ vertices contains $\frac{1}{8+\sqrt{32}}n^2(1-o(1))$ arc-disjoint directed 4-cycles. The result is also used to provide an upper bound for the distance between two antipodal vertices in interchange graphs.
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。
Towards Transitive-free Digraphs
本文研究 $\ell$-RTVD 问题，在多类有向图上给出多项式可解、NP 难及参数化核与不可处理性结果。
Let $U_5$ be the tournament with vertices $v_1$, ..., $v_5$ such that $v_2 \rightarrow v_1$, and $v_i \rightarrow v_j$ if $j-i \equiv 1$, $2 \pmod{5}$ and ${i,j} \neq {1,2}$. In this paper we describe the tournaments which do not have $U_5$ as a subtournament. Specifically, we show that if a tournament $G$ is "prime"---that is, if there is no subset $X \subseteq V(G)$, $1 < |X| < |V(G)|$, such that for all $v \in V(G) \backslash X$, either $v \rightarrow x$ for all $x \in X$ or $x \rightarrow v$ for all $x \in X$---then $G$ is $U_5$-free if and only if either $G$ is a specific tournament $T_n$…
On heroes in digraphs with forbidden induced forests
本文研究禁止诱导森林的有向图类的英雄，刻画了不同度定向星及禁止定向扫帚情形下能保证二色数有界的英雄结构。
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 $n$.
In this paper, we give a direct construction for a set of dice realizing any given tournament $T$. The construction for a tournament with $n$ vertices requires a number of sides on the order of $n$, which appears to be the best general construction to date. Our construction relies only on a standard theorem from graph theory.
A digraph $D$ is $k$-linked if it satisfies that for every choice of disjoint sets $\{x_1,\ldots{},x_k\}$ and $\{y_1,\ldots{},y_k\}$ of vertices of $D$ there are vertex disjoint paths $P_1,\ldots{},P_k$ such that $P_i$ is an $(x_i,y_i)$-path. Confirming a conjecture by Kühn et al, Pokrovskiy proved in 2015 that every $452k$-strong tournament is $k$-linked and asked for a better linear bound. Very recently Meng et al proved that every $(40k-31)$-strong tournament is $k$-linked. In this note we use an important lemma from their paper to give a short proof that every $(13k-6)$-strong tournament o…
Immersions of directed graphs in tournaments
本文研究锦标赛中有向图的浸入问题，证明了任意具有O(k)个顶点的锦标赛必含k个顶点的传递锦标赛的1-浸入，并进一步证明最小出度至少为Ck的锦标赛必含k个顶点的完全有向图的2-浸入，且这些界在相应意义下都是紧的。
Given a tournament $T$, the problem MaxCT consists of finding a maximum (arc-disjoint) cycle packing of $T$. In the same way, MaxTT corresponds to the specific case where the collection of cycles are triangles (i.e. directed 3-cycles). Although MaxCT can be seen as the LP dual of minimum feedback arc set in tournaments which have been widely studied, surprisingly no algorithmic results seem to exist concerning the former. In this paper, we prove the NP-hardness of both MaxCT and MaxTT. We also show that deciding if a tournament has a cycle packing and a feedback arc set with the same size is a…
Tournament transitivity of graphs
本文研究图的竞赛传递性问题，证明其判定版本在弦图等上是NP完全的、在树上可多项式时间求解，并刻画了三类BCG中图的传递性与竞赛传递性的相等性情况。
We study a high-dimensional analog for the notion of an acyclic (aka transitive) tournament. We give upper and lower bounds on the number of $d$-dimensional $n$-vertex acyclic tournaments. In addition, we prove that every $n$-vertex $d$-dimensional tournament contains an acyclic subtournament of $Ω(\log^{1/d}n)$ vertices and the bound is tight. This statement for tournaments (i.e., the case $d=1$) is a well-known fact. We indicate a connection between acyclic high-dimensional tournaments and Ramsey numbers of hypergraphs. We investigate as well the inter-relations among various other notions o…
The inversion number of a path-reversed tournament: Resolving a conjecture of Belkhechine, Bouaziz, Boudabbous, and Pouzet
本文解决了关于路径反转竞赛图逆序数的猜想，该竞赛图由自然传递竞赛图反转特定连续对得到，猜想自然路径反转族的逆序数为\(\left\lfloor(n - 1)/2\right\rfloor\)，通过相关研究得出了结论。
The Hajnal--Szemerédi theorem states that for any integer $r \ge 1$ and any multiple $n$ of $r$, if $G$ is a graph on $n$ vertices and $δ(G) \ge (1 - 1/r)n$, then $G$ can be partitioned into $n/r$ vertex-disjoint copies of the complete graph on $r$ vertices. We prove a very general analogue of this result for directed graphs: for any integer $r \ge 4$ and any sufficiently large multiple $n$ of $r$, if $G$ is a directed graph on $n$ vertices and every vertex is incident to at least $2(1 - 1/r)n - 1$ directed edges, then $G$ can be partitioned into $n/r$ vertex-disjoint subgraphs of size $r$ eac…
We investigate the \emph{last passage percolation} problem on transitive tournaments, in the case when the edge weights are independent Bernoulli random variables. Given a transitive tournament on $n$ nodes with random weights on its edges, the last passage percolation problem seeks to find the weight $X_n$ of the heaviest path, where the weight of a path is the sum of the weights on its edges. We give a recurrence relation and use it to obtain a (bivariate) generating function for the probability generating function of $X_n$. This also gives exact combinatorial expressions for $\mathbb{E}[X_n…
A $k$-majority digraph is a directed graph created by combining $k$ individual rankings on the same ground set to form a consensus where edges point in the direction indicated by a strict majority of the rankings. The $k$-majority digraph is used to model voting scenarios, where the vertices correspond to options ranked by $k$ voters. When $k$ is odd, the resulting digraph is always a tournament, called $k$-majority tournament. Let $f_k(n)$ be the minimum, over all $k$-majority tournaments with $n$ vertices, of the maximum order of an induced transitive sub-tournament. Recently, Milans, Schrei…
Let S_m denote the m-vertex simple digraph formed by m-1 edges with a common tail. Let f(m) denote the minimum n such that every n-vertex tournament has a spanning subgraph consisting of n/m disjoint copies of S_m. We prove that m lg m - m lg lg m <= f(m) <= 4m^2 - 6m for sufficiently large m.
We propose a purely combinatorial quadratic time algorithm that for any $n$-vertex $P_{k}$-free tournament $T$, where $P_{k}$ is a directed path of length $k$, finds in $T$ a transitive subset of order $n^{\frac{c}{k\log(k)^{2}}}$. As a byproduct of our method, we obtain subcubic $O(n^{1-\frac{c}{k\log(k)^{2}}})$-approximation algorithm for the optimal acyclic coloring problem on $P_{k}$-free tournaments. Our results are tight up to the $\log(k)$-factor in the following sense: there exist infinite families of $P_{k}$-free tournaments with largest transitive subsets of order at most $n^{\frac{c…
Tournaments are orientations of the complete graph, and the directed Ramsey number $R(k)$ is the minimum number of vertices a tournament must have to be guaranteed to contain a transitive subtournament of size $k$, which we denote by $TT_k$. We include a computer-assisted proof of a conjecture by Sanchez-Flores that all $TT_6$-free tournaments on 24 and 25 vertices are subtournaments of $ST_{27}$, the unique largest TT_6-free tournament. We also classify all $TT_6$-free tournaments on 23 vertices. We use these results, combined with assistance from SAT technology, to obtain the following impro…
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 \emph{chromatic number} of a directed graph $D$ is the minimum number of colors needed to color the vertices of $D$ such that each color class of $D$ induces an acyclic subdigraph. Thus, the chromatic number of a tournament $T$ is the minimum number of transitive subtournaments which cover the vertex set of $T$. We show in this paper that tournaments are significantly simpler than graphs with respect to coloring. Indeed, while undirected graphs can be altogether "locally simple" (every neighborhood is a stable set) and have large chromatic number, we show that locally simple tournaments ar…
Some results and problems on tournament structure
本文综述了关于大色数竞赛图中是否存在两个顶点不相交且边单向指向的子竞赛图的问题，并报告了在竞赛图着色与支配相关问题上的研究进展。
The Rényi $α$-entropy $H_α$ of complete antisymmetric directed graphs (i.e., tournaments) is explored. We optimize $H_α$ when $α= 2$ and $3$, and find that as $α$ increases $H_α$'s sensitivity to what we refer to as `regularity' increases as well. A regular tournament on $n$ vertices is one with each vertex having out-degree $\frac{n-1}{2}$, but there is a lot of diversity in terms of structure among the regular tournaments; for example, a regular tournament may be such that each vertex's out-set induces a regular tournament (a doubly-regular tournament) or a transitive tournament (a rotationa…
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