On the Caccetta-Haggkvist conjecture with forbidden subgraphs

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 …

Properly colored short cycles in edge-colored graphs

Properly colored cycles in edge-colored graphs are closely related to directed cycles in oriented graphs. As an analogy of the well-known Caccetta-Häggkvist Conjecture, we study the existence of properly colored cycles of bounded length in an edge-colored graph. We first prove that for all integers $s$ and $t$ with $t\geq s\geq2$, every edge-colored graph $G$ with no properly colored $K_{s,t}$ contains a spanning subgraph $H$ which admits an orientation $D$ such that every directed cycle in $D$ is a properly colored cycle in $G$. Using this result, we show that for $r\geq4$, if the Caccetta-Hä…

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。

迫使有向圈存在的度条件

Degree conditions forcing directed cycles

本文研究迫使有向图含指定有向圈的最小半度阈值，解决了 Kelly、Kühn 和 Osthus 关于长度被 3 整除且非 3 的圈的猜想，并给出给定方向圈的渐近阈值。

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。

Notes on Aharoni's rainbow cycle conjecture

In 2017, Ron Aharoni made the following conjecture about rainbow cycles in edge-coloured graphs: If $G$ is an $n$-vertex graph whose edges are coloured with $n$ colours and each colour class has size at least $r$, then $G$ contains a rainbow cycle of length at most $\lceil \frac{n}{r} \rceil$. One motivation for studying Aharoni's conjecture is that it is a strengthening of the Caccetta-Häggkvist conjecture on digraphs from 1978. In this article, we present a survey of Aharoni's conjecture, including many recent partial results and related conjectures. We also present two new results. Our main…

Aharoni的彩虹圈猜想在相差一个加性常数的意义下成立

Aharoni's rainbow cycle conjecture holds up to an additive constant

本研究针对Aharoni提出的彩虹圈猜想，证明对于固定r，该猜想在相差一个O(r^5 log²r)阶加性常数的意义下成立，推进了边着色图中彩虹圈存在性问题的研究。

匹配族的彩虹圈

Rainbow cycles for families of matchings

本文研究边着色图的彩虹周长，推广了Caccetta-Häggkvist猜想，并证明当色类为大小至少为2的匹配时，$n$个匹配的彩虹周长为$O(\log n)$。

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。

边着色图中的短彩虹圈

Short rainbow cycles in edge-colored graphs

本文扩展了Guo关于Aharoni彩虹圈猜想的研究，证明了当边着色图中超过半数的颜色类为大小为2的匹配或三角形时，必然存在长度为 $O(\log n)$ 的短彩虹圈，并验证了该对数界的数量级正确性。