Directed Triangles in Digraphs

AbstractLetcbe the smallest possible value such that every digraph onnvertices with minimum outdegree at leastcncontains a directed triangle. It was conjectured by Caccetta and Häggkvist in 1978 thatc=1/3. Recently Bondy showed thatc⩽(26−3)/5=0.3797… by using some counting arguments. In this note, we prove thatc⩽3−7=0.3542…

Short rainbow cycles in graphs and matroids

Let $G$ be a simple $n$-vertex graph and $c$ be a colouring of $E(G)$ with $n$ colours, where each colour class has size at least $2$. We prove that $(G,c)$ contains a rainbow cycle of length at most $\lceil \frac{n}{2} \rceil$, which is best possible. Our result settles a special case of a strengthening of the Caccetta-H\"aggkvist conjecture, due to Aharoni. We also show that the matroid generalization of our main result also holds for cographic matroids, but fails for binary matroids.Comment: 9 pages, 2 figure

A note on directed 4-cycles in digraphs

Using some combinatorial techniques, in this note, it is proved that if $\alpha\geq 0.28866$, then any digraph on $n$ vertices with minimum outdegree at least $\alpha n$ contains a directed cycle of length at most 4

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.Comment: 19 pages, 7 figures; this is the final version to appear in Combinatoric

Directed triangles in directed graphs

AbstractWe show that each directed graph (with no parallel arcs) on n vertices, each with indegree and outdegree at least n/t where t=2.888997… contains a directed circuit of length at most 3