Jan Hladky, Daniel Kral, Sergey Norin
Motivated by the Caccetta-Haggkvist Conjecture, we prove that every digraph on n vertices with minimum outdegree 0.3465n contains an oriented triangle. This improves the bound of 0.3532n of Hamburger, Haxell and Kostochka. The main new tool we use in our proof is the theory of flag algebras developed recently by Razborov.
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In this work, we establish an analogue result of the Erdös-Stone theorem of weighted digraphs using Regularity Lemma of digraphs. We give a stability result of oriented graphs and digraphs with forbidden blow-up transitive triangle and show that almost all oriented graphs and almost all digraphs with forbidden blow-up transitive triangle are almost bipartite respectively.
Let $D$ be an arc-colored digraph. The arc number $a(D)$ of $D$ is defined as the number of arcs of $D$. The color number $c(D)$ of $D$ is defined as the number of colors assigned to the arcs of $D$. A rainbow triangle in $D$ is a directed triangle in which every pair of arcs have distinct colors. Let $f(D)$ be the smallest integer such that if $c(D)\geq f(D)$, then $D$ contains a rainbow triangle. In this paper we obtain $f(\overleftrightarrow{K}_{n})$ and $f(T_n)$, where $\overleftrightarrow{K}_{n}$ is a complete digraph of order $n$ and $T_n$ is a strongly connected tournament of order $n$.…
For a simple digraph $G$ without directed triangles or digons, let $β(G)$ be the size of the smallest subset $X \subseteq E(G)$ such that $G\setminus X$ has no directed cycles, and let $γ(G)$ be the number of unordered pairs of nonadjacent vertices in $G$. In 2008, Chudnovsky, Seymour, and Sullivan showed that $β(G) \le γ(G)$, and conjectured that $β(G) \le γ(G)/2$. Recently, Dunkum, Hamburger, and Pór proved that $β(G) \le 0.88 γ(G)$. In this note, we prove that $β(G) \le 0.8616 γ(G)$.
Comparability graphs are a popular class of graphs. We introduce as the digraph analogue of comparability graphs the class of comparability digraphs. We show that many concepts such as implication classes and the knotting graph for a comparability graph can be naturally extended to a comparability digraph. We give a characterization of comparability digraphs in terms of their knotting graphs. Semicomplete comparability digraphs are a prototype of comparability digraphs. One instrumental technique for analyzing the structure of comparability graphs is the Triangle Lemma for graphs. We generaliz…
Ramsey-Turán Type Problem for Perfect Transitive Triangle Tilings in Digraphs
研究有向图在独立数较小条件下，通过最小度条件保证完美传递三角形铺砌的存在性，给出了渐近最优的最小度下界。
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…
In this paper, we investigate the relation between the (fractional) domination number of a digraph $G$ and the independence number of its underlying graph, denoted by $α(G)$. More precisely, we prove that every digraph $G$ has fractional domination number at most $2α(G)$, and every directed triangle-free digraph $G$ has domination number at most $α(G)\cdot α(G)!$. The first bound is sharp.
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 …
The dichromatic number of an oriented graph is the minimum size of a partition of its vertices into acyclic induced subdigraphs. We prove that oriented graphs with no induced directed path on six vertices and no triangle have bounded dichromatic number. This is one (small) step towards the general conjecture asserting that for every oriented tree T and every integer k, any oriented graph that does not contain an induced copy of T nor a clique of size k has dichromatic number at most some function of k and T.
Assume that D is a digraph without cyclic triangles and its vertices are partitioned into classes A_1,...,A_t of independent vertices. A set $U=\cup_{i\in S} A_i$ is called a dominating set of size |S| if for any vertex $v\in \cup_{i\notin S} A_i$ there is a w in U such that (w,v) is in E(D). Let beta(D) be the cardinality of the largest independent set of D whose vertices are from different partite classes of D. Our main result says that there exists a h=h(beta(D)) such that D has a dominating set of size at most h. This result is applied to settle a problem related to generalized Gallai colo…
Let $\mathcal{G}$ be a properly face 2-coloured (say black and white) \break piecewise-linear triangulation of the sphere with vertex set $V$. Consider the abelian group $\mathcal{A}_W$ generated by the set $V$, with relations $r+c+s=0$ for all white triangles with vertices $r$, $c$ and $s$. The group $\mathcal{A}_B$ can be defined similarly, using black triangles. These groups are related in the following manner $\mathcal{A}_W\cong\mathcal{A}_B\cong\mathbb{Z}\oplus\mathbb{Z}\oplus\mathcal{C}$ where $\mathcal{C}$ is a finite abelian group. The finite torsion subgroup $\mathcal{C}$ is referred …
A digraph is connected-homogeneous if any isomorphism between finite connected induced subdigraphs extends to an automorphism of the digraph. We consider locally-finite connected-homogeneous digraphs with more than one end. In the case that the digraph embeds a triangle we give a complete classification, obtaining a family of tree-like graphs constructed by gluing together directed triangles. In the triangle-free case we show that these digraphs are highly arc-transitive. We give a classification in the two-ended case, showing that all examples arise from a simple construction given by gluing …
Combinatorial Games and the Golden Ratio on Digraphs
本文引入了名为三角游戏的新型组合游戏，探讨了有向3环图上的策略与misère玩法，并揭示黄金分割比在游戏分析中的关键作用。
The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has been a recent center of study. In this work we look at possible extensions of Gyárfás-Sumner conjecture. More precisely, we propose as a conjecture a simple characterization of finite sets $\mathcal F$ of digraphs such that every oriented graph with sufficiently large dichromatic number must contain a member of $\mathcal F$ as an induce subdigraph. Among notable…
Using Razborov's flag algebras we show that a triangle-free graph on n vertices contains at most (n/5)^5 cycles of length five. It settles in the affirmative a conjecture of Erdos.
We prove that for every $n$, there is a graph $G$ with $χ(G) \geq n$ and $ω(G) \leq 3$ such that every induced subgraph $H$ of $G$ with $ω(H) \leq 2$ satisfies $χ(H) \leq 4$. This disproves a well-known conjecture. Our construction is a digraph with bounded clique number, large dichromatic number, and no induced directed cycles of odd length at least 5.
A famous conjecture of Caccetta and Häggkvist is that in a digraph on $n$ vertices and minimum out-degree at least $\frac{n}{r}$ there is a directed cycle of length $r$ or less. We consider the following generalization: in an undirected graph on $n$ vertices, any collection of $n$ disjoint sets of edges, each of size at least $\frac{n}{r}$, has a rainbow cycle of length $r$ or less. We focus on the case $r=3$, and prove the existence of a rainbow triangle under somewhat stronger conditions than in the conjecture. For any fixed $k$ and large enough $n$, we determine the maximum number of edges …
A star-comb lemma for infinite digraphs
本研究将无限图论中的星-梳引理拓展至有向图领域，证明强连通有向图的任意无限顶点集对应存在含无限多相关顶点的强连通蝴蝶子式，形状为星、梳或三角形链。
The \textit{acyclic disconnection} $\overrightarrow{ω}(D)$ (resp. the \textit{directed triangle free disconnection } $\overrightarrow{ω}_{3}(D)$) of a digraph $D$ is defined as the maximum possible number of connected components of the underlying graph of $D\setminus A(D^{\ast })$ where $D^{\ast }$ is an acyclic (resp. a directed triangle free) subdigraph of $D$. In this paper, we generalize some previous results and solve some problems posed by V. Neumann-Lara (The acyclic disconnection of a digraph, Discrete Math. 197/198 (1999), 617-632). Let $\overrightarrow{C}_{2n+1}(J)$ be a circulant to…
Density of rainbow triangles and properly colored $K_4$'s
本文利用无计算机的旗代数方法，给出了具有红、绿、蓝边数的图中正确3边着色 $K_4$ 数量的尖锐上界，并推广到任意完全图 $K_k$ 的固定彩虹着色，同时提供了经典计数和熵方法的证明。
Using the formalism of flag algebras, we prove that every triangle-free graph $G$ with $n$ vertices contains at most $(n/5)^5$ cycles of length five. Moreover, the equality is attained only when $n$ is divisible by five and $G$ is the balanced blow-up of the pentagon. We also compute the maximal number of pentagons and characterize extremal graphs in the non-divisible case provided $n$ is sufficiently large. This settles a conjecture made by Erdős in 1984.
Erdos and Sos proposed a problem of determining the maximum number F(n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that F(n) = F(a)+ F(b)+F(c)+F(d)+abc+abd+acd+bcd, where a+b+c+d = n and a, b, c, d are as equal as possible. We prove that the conjectured recurrence holds for sufficiently large n. We also prove the conjecture for n = 4k for all k. These results imply that lim F(n) n^3/6 = 0.4, and determine the unique limit object. In the proof we use flag algebras combined with stability arguments.
We say that a (di)graph $G$ has a perfect $H$-packing if there exists a set of vertex-disjoint copies of $H$ which cover all the vertices in $G$. The seminal Hajnal--Szemerédi theorem characterises the minimum degree that ensures a graph $G$ contains a perfect $K_r$-packing. In this paper we prove the following analogue for directed graphs: Suppose that $T$ is a tournament on $r$ vertices and $G$ is a digraph of sufficiently large order $n$ where $r$ divides $n$. If $G$ has minimum in- and outdegree at least $ (1-1/r)n$ then $G$ contains a perfect $T$-packing. In the case when $T$ is a cyclic …
Improved bounds for the triangle case of Aharoni's rainbow generalization of the Caccetta-Häggkvist conjecture
本文改进了Aharoni关于Caccetta-Häggkvist猜想彩虹推广的三角形式界，证明了(1.1077,1/3)、(1,0.3988)及(1.3481,1/4)等参数对是三角的。
Algorithms and hardness for Metric Dimension on digraphs
研究有向图上的度量维数问题，提出线性时间算法适用于树结构下的有向图，并扩展至单环图的定向结构，同时给出基于有向模宽参数的固定参数可解算法，证明该问题在平面无三角形无环有向图上为NP难。
Let $G$ be an $n$-vertex oriented graph. Let $t(G)$ (respectively $i(G)$) be the probability that a random set of $3$ vertices of $G$ spans a transitive triangle (respectively an independent set). We prove that $t(G) + i(G) \geq \frac{1}{9}-o_n(1)$. Our proof uses the method of flag algebras that we supplement with several steps that make it more easily comprehensible. We also prove a stability result and an exact result. Namely, we describe an extremal construction, prove that it is essentially unique, and prove that if $H$ is sufficiently far from that construction, then $t(H) + i(H)$ is sig…
An orientation of $G$ is a digraph obtained from $G$ by replacing each edge by exactly one of two possible arcs with the same endpoints. We call an orientation \emph{proper} if neighbouring vertices have different in-degrees. The proper orientation number of a graph $G$, denoted by $\vecχ(G)$, is the minimum maximum in-degree of a proper orientation of G. Araujo et al. (Theor. Comput. Sci. 639 (2016) 14--25) asked whether there is a constant $c$ such that $\vecχ(G)\leq c$ for every outerplanar graph $G$ and showed that $\vecχ(G)\leq 7$ for every cactus $G.$ We prove that $\vecχ(G)\leq 3$ if $G…
We introduce the following combinatorial problem. Let $G$ be a triangle-free regular graph with edge density $ρ$. What is the minimum value $a(ρ)$ for which there always exist two non-adjacent vertices such that the density of their common neighborhood is $\leq a(ρ)$? We prove a variety of upper bounds on the function $a(ρ)$ that are tight for the values $ρ=2/5,\ 5/16,\ 3/10,\ 11/50$, with $C_5$, Clebsch, Petersen and Higman-Sims being respective extremal configurations. Our proofs are entirely combinatorial and are largely based on counting densities in the style of flag algebras. For small v…
The least signless Laplacian eigenvalue of $\{C_3,C_5\}$-free graphs
本文研究无三角形且无五边形的图的无符号拉普拉斯最小特征值，证明其小于 $0.0569n$，无需正则性假设，方法结合旗不等式与局部 Rayleigh 约束。
Geometries with trialities arising from linear spaces
本文提出通过任意秩二关联系统的三角形复形构造具有三重性的秩三关联系统，并证明当底系统为线性空间时，可得到首个具有三重性但无对偶性的厚、旗传递且剩余连通的几何无限族。
Towards an edge-coloured Corrádi--Hajnal theorem
本研究将经典Corrádi-Hajnal定理推广到边着色图，证明最小色度不小于(5/6+ε)n的n顶点边着色图存在生成顶点不交彩虹三角形填充，同时给出下界至少为5n/7的构造，并讨论了有向图相关填充问题。
The Erdős Pentagon problem asks to find an $n$-vertex triangle-free graph that is maximizing the number of $5$-cycles. The problem was solved using flag algebras by Grzesik and independently by Hatami, Hladký, Král', Norin, and Razborov. Recently, Palmer suggested the general problem of maximizing the number of $5$-cycles in $K_{k+1}$-free graphs. Using flag algebras, we show that every $K_{k+1}$-free graph of order $n$ contains at most \[\frac{1}{10k^4}(k^4 - 5k^3 + 10k^2 - 10k + 4)n^5 + o(n^5)\] copies of $C_5$ for any $k \geq 3$, with the Turán graph begin the extremal graph for large enoug…
The Spectrum of Triangle-free Graphs
研究三角自由图的谱性质，证明其符号拉普拉斯矩阵的最小特征值上界，并与Erdős的两个猜想相关联。
A sharp bound for the least signless Laplacian eigenvalue of triangle-free graphs
该研究证明了顶点数$n\geq1$的有限简单无三角形图的最小无符号拉普拉斯特征值的紧界$7n/50$，证实了相关猜想，采用含局部瑞利不等式与flag-algebra半定规划的方法完成证明。
The guessing game introduced by Riis is a variant of the "guessing your own hats" game and can be played on any simple directed graph G on n vertices. For each digraph G, it is proved that there exists a unique guessing number gn(G) associated to the guessing game played on G. When we consider the directed edge to be bidirected, in other words, the graph G is undirected, Christofides and Markstrom introduced a method to bound the value of the guessing number from below using the fractional clique number Kf(G). In particular they showed gn(G) >= |V(G)| - Kf(G). Moreover, it is pointed out that…
A Higgs category for the cluster variety of triples of flags
本文为Dynkin类型Δ的旗三元组的簇变种构造并研究了一个Higgs范畴（Yilin Wu意义下），该范畴是Frobenius且稳定2-Calabi-Yau的，并具有预期的循环群和辫群对称性，其构造依赖于Merlin Christ的一个猜想。
Minimum acyclic number and maximum dichromatic number of oriented triangle-free graphs of a given order
本文研究阶数为 $n$ 的定向无三角形图的最小无环数与最大二色数，给出二者的渐近上下界，并确定小阶情形下二色数为 3 的最小构造为 25 个顶点。
On Seymour's and Sullivan's Second Neighbourhood Conjectures
研究西摩和沙利文关于有向图顶点邻域关系的第二邻域猜想，给出满足沙利文猜想的传递三角形数量充分条件，证明其对平面图等成立，还表明对一些定向分裂图族两个猜想也成立。
Towards Esperet's Conjecture: Polynomial $χ$-Bounds for Structured Graph Classes
本文研究了结构图类的色数界，证明了特定图类在排除某些子图时色数具有多项式或线性界。
The Erdos n^2/25 max-cut conjecture for small multiples of five, via a per-root-MaxCut envelope and blow-up integrality
通过计算机辅助证明，结合每根最大割包络、旗代数证书和爆破整性，证明了 Erdős 猜想对于 N=5n (n≤40) 成立，即 a(5n)=n^2。
The codegree threshold $\mathrm{ex}_2(n, F)$ of a $3$-graph $F$ is the minimum $d=d(n)$ such that every $3$-graph on $n$ vertices in which every pair of vertices is contained in at least $d+1$ edges contains a copy of $F$ as a subgraph. We study $\mathrm{ex}_2(n, F)$ when $F=K_4^-$, the $3$-graph on $4$ vertices with $3$ edges. Using flag algebra techniques, we prove that if $n$ is sufficiently large then $\mathrm{ex}_2(n, K_4^-)\leq (n+1)/4$. This settles in the affirmative a conjecture of Nagle from 1999. In addition, we obtain a stability result: for every near-extremal configuration $G$, t…
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