Proof of the Caccetta-Haggkvist conjecture for digraphs with small independence number

非均匀度与Caccetta-Häggkvist猜想的彩虹版本

Non-uniform degrees and rainbow versions of the Caccetta-Häggkvist conjecture

针对Caccetta-Häggkvist猜想，提出基于所有出度的有向围长上界，进而提出彩虹环版本的猜想，并在边集规模为1-2及边集为三角形的情形下取得验证结果。

Aharoni 对 Caccetta-Häggkvist 猜想的彩虹推广的进一步逼近

Further approximations for Aharoni's rainbow generalization of the Caccetta-Häggkvist conjecture

本文研究 Aharoni 对 Caccetta-Häggkvist 猜想的彩虹推广，将每个颜色类大小的下界从 Ω(k log k) 改进为 Ω(k)，证明此时仍存在长度至多 ⌈n/k⌉ 的彩虹环，并进一步讨论了颜色数量条件放宽后的情形。

Aharoni的Caccetta-Häggkvist猜想彩虹推广三角形式的改进界

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。

Strong connectivity and directed triangles in oriented graphs. Partial results on a particular case of the Caccetta-Häggkvist conjecture

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 …

Short directed cycles in bipartite digraphs

The Caccetta-Häggkvist conjecture implies that for every integer $k\ge 1$, if $G$ is a bipartite digraph, with $n$ vertices in each part, and every vertex has out-degree more than $n/(k+1)$, then $G$ has a directed cycle of length at most $2k$. If true this is best possible, and we prove this for $k = 1,2,3,4,6$ and all $k\ge 224,539$. More generally, we conjecture that for every integer $k\ge 1$, and every pair of reals $α, β> 0$ with $kα+β>1$, if $G$ is a bipartite digraph with bipartition $(A,B)$, where every vertex in $A$ has out-degree at least $β|B|$, and every vertex in $B$ has out-degr…

Counting flags in triangle-free digraphs

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.

匹配和三角形的族的短彩虹圈

Short rainbow cycles for families of matchings and triangles

本文研究了边集族中彩虹圈的长度问题，证明了在混合包含大小为2的匹配或三角形的情况下，彩虹圈长度渐近为O(log n)，并确定了引入单边时彩虹周长从线性变为对数的阈值比例。

On the size of spheres of relations with a transitive group of automorphisms

Let $Γ=(V,E)$ be a point-transitive reflexive relation. Let $v\in V$ and put $r=|Γ(v)|.$ Also assume $Γ^j(v)\cap Γ^{-}(v)=\{v\}$. Then $$ |Γ^{j} (v)\setminus Γ^{j-1} (v)| \ge r-1.$$ In particular we have $ |Γ^{j} (v)| \ge 1+(r-1)j.$ The last result confirms a recent conjecture of Seymour in the case vertex-transitive graphs. Also it gives a short proof for the validity of the Caccetta-Häggkvist conjecture for vertex-transitive graphs and generalizes an additive result of Shepherdson.

Arbitrary Orientations of Hamilton Cycles in Digraphs

Let $n$ be sufficiently large and suppose that $G$ is a digraph on $n$ vertices where every vertex has in- and outdegree at least $n/2$. We show that $G$ contains every orientation of a Hamilton cycle except, possibly, the antidirected one. The antidirected case was settled by DeBiasio and Molla, where the threshold is $n/2+1$. Our result is best possible and improves on an approximate result by Häggkvist and Thomason.

On iterated image size for point-symmetric relations

Let $Γ=(V,E)$ be a point-symmetric reflexive relation and let $v\in V$ such that $|Γ(v)|$ is finite (and hence $|Γ(x)|$ is finite for all $x$, by the transitive action of the group of automorphisms). Let $j\in \N$ be an integer such that $Γ^j(v)\cap Γ^{-}(v)=\{v\}$. Our main result states that $$ |Γ^{j} (v)|\ge | Γ^{j-1} (v)| + |Γ(v)|-1.$$ As an application we have $ |Γ^{j} (v)| \ge 1+(|Γ(v)|-1)j.$ The last result confirms a recent conjecture of Seymour in the case of vertex-symmetric graphs. Also it gives a short proof for the validity of the Caccetta-Häggkvist conjecture for vertex-symmetric…

关于 $k$-反传递与 $m$-自由有向图的第二邻域问题的一个注记

A note on the second neighborhood problem for $k$-anti-transitive and $m$-free digraphs

本文证明：若有限有向图 $D$ 是 $k$-反传递且 $(k-4)$-自由的，则 $D$ 必存在 Seymour 顶点，从而在 7-反传递定向图上验证了 Caccetta-Haggkvist 猜想的一个特例，扩展了已知结果。

Outdegree conditions forcing short cycles in digraphs

Given a positive integer $m\ge 3$, let $ch(m)$ be the smallest positive constant with the following property: \emph{ Every simple directed graph on $n\ge 3$ vertices all whose outdegrees are at least $ch(m)\cdot n$ contains a directed cycle of length at most $m$.} Caccetta and Häggkvist conjectured that $ch(m)=1/m$, which if true, would be the best possible. In this paper, we prove the following result: \emph{ For every integer $m\ge 3$, let $α(m)$ be the unique real root in $(0,1)$ of the equation} \begin{equation*} (1-x)^{m-2}=\frac{3x}{2-x}. \end{equation*} Then $ch(m)\le α(m)$. This genera…

关于有向图中定向路径的长度

On the length of directed paths in digraphs

本文推广了 Thomassé 猜想的反例，证明其对所有围长 g≥4 均不成立，并给出了有向图定向路径长度的多个非平凡下界，包括一般情形和正则图情形。

公开不交环与正则有向图的定向树宽

Openly disjoint cycles and directed tree-width of regular digraphs

研究正则有向图中公开不交环的数量及其与定向树宽的关系，证明正则有向图的公开不交环数下界与上界，推动有向图子图分解的度数条件研究。