The Caccetta-Haggkvist conjecture states that if G is a finite directed graph with at least n/k edges going out of each vertex, then G contains a directed cycle of length at most k. Hamidoune used methods and results from additive number theory to prove the conjecture for Cayley graphs and for vertex-transitive graphs. This expository paper contains a survey of results on the Caccetta-Haggkvist conjecture, and complete proofs of the conjecture in the case of Cayley and vertex-transitive graphs.
This paper is an attempt to survey the current state of our knowledge on the Caccetta-Haggkvist conjecture and related questions. In January 2006 there was a workshop hosted by the American Institute of Mathematics in Palo Alto, on the Caccetta-Haggkvist conjecture, and this paper partly originated there, as a summary of the open problems and partial results presented at the workshop. This summary includes results and open problems related to Caccetta-Haggkvist, Seymour's Second Neighborhood Conjecture, the k/2 Conjecture (for nonedges), and connections with algebraic number theory through Cay…
For a digraph $G$ and $v \in V(G)$, let $δ^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-Häggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $δ^+(v) \ge n/k$ for all $v \in V(G)$, then G contains a directed cycle of length at most $k$. In [2], N. Lichiardopol proved that this conjecture is true for digraphs with independence number equal to two. In this paper, we generalize that result, proving that the conjecture is true for digraphs with independence number at most $(k+1)/2$.
The Caccetta-Haggkvist conjecture made in 1978 asserts that every orgraph on n vertices without oriented cycles of length <= l must contain a vertex of outdegree at most (n-1)/l. It has a rather elaborate set of (conjectured) extremal configurations. In this paper we consider the case l=3 that received quite a significant attention in the literature. We identify three orgraphs on four vertices each that are missing as an induced subgraph in all known extremal examples and prove the Caccetta-Haggkvist conjecture for orgraphs missing as induced subgraphs any of these orgraphs, along with cycles …
In the theory of digraphs, the study of cycles is a subject of great importance and has given birth to a number of deep questions such as the Behzad-Chartrand-Wall conjecture (1970) and its generalization, the Caccetta-Häggkvist conjecture (1978). Despite a lot of interest and efforts, the progress on these remains slow and mostly restricted to the solution of some special cases. In this note, we prove these conjectures for digraphs with girth is at least as large as their minimum out-degree and without short even cycles. More generally, we prove that if a digraph has sufficiently large girth …
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 …
For a digraph $G$ and $v \in V(G)$, let $δ^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-Häggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $δ^+(v) \ge k$ for all $v \in V(G)$, then G contains a directed cycle of length at most $\lceil n/k \rceil$. In [2], Aharoni proposes a generalization of this conjecture, that a simple edge-colored graph on $n$ vertices with $n$ color classes, each of size $k$, has a rainbow cycle of length at most $\lceil n/k \rceil$. In this paper, we prove this conjecture if each color class has size $Ω…
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 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…
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…
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.
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.
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…
A graph is called diameter-$k$-critical if its diameter is $k$, and the removal of any edge strictly increases the diameter. In this paper, we prove several results related to a conjecture often attributed to Murty and Simon, regarding the maximum number of edges that any diameter-$k$-critical graph can have. In particular, we disprove a longstanding conjecture of Caccetta and Häggkvist (that in every diameter-2-critical graph, the average edge-degree is at most the number of vertices), which promised to completely solve the extremal problem for diameter-2-critical graphs. On the other hand, w…
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…
Simon Fraser University（西蒙菲莎大学）；Université Libre de Bruxelles（布鲁塞尔自由大学）；Douglas College（道格拉斯学院）；Monash University（莫纳什大学）；University of Ljubljana（卢布尔雅那大学）；UNAM campus Juriquilla（墨西哥国立自治大学希奎拉校区）
