Abstract
The Caccetta-Häggkvist Conjecture asserts that every oriented graph on $n$ vertices without directed cycles of length less than or equal to $l$ has minimum outdegree at most $(n-1)/l$. In this paper we state a conjecture for graphs missing a transitive tournament on $2^k+1$ vertices, with a weaker assumption on minimum outdegree. We prove that the Caccetta-Häggkvist Conjecture follows from the presented conjecture and show matching constructions for all $k$ and $l$. The main advantage of considering this generalized conjecture is that it reduces the set of the extremal graphs and allows using an induction.We also prove the triangle case of the conjecture for $k=1$ and $2$ by using the Razborov's flag algebras. In particular, it proves the most interesting and studied case of the Caccetta-Häggkvist Conjecture in the class of graphs without the transitive tournament on 5 vertices. It is also shown that the extremal graph for the case $k=2$ has to be a blow-up of a directed cycle on 4 vertices having in each blob an extremal graph for the case $k=1$ (complete regular bipartite graph), which confirms the conjectured structure of the extremal examples.
Highlights
We prove the triangle case of the conjecture for k = 1 and 2 by using the Razborov’s flag algebras
In this paper we state a conjecture for graphs missing a transitive tournament on 2k + 1 vertices, with a weaker assumption on minimum outdegree
It is shown that the extremal graph for the case k = 2 has to be a blow-up of a directed cycle on 4 vertices having in each blob an extremal graph for the case k = 1, which confirms the conjectured structure of the extremal examples
Summary
The Caccetta-Haggkvist Conjecture asserts that every oriented graph on n vertices without directed cycles of length less than or equal to l has minimum outdegree at most (n − 1)/l. It proves the most interesting and studied case of the Caccetta-Haggkvist Conjecture in the class of graphs without the transitive tournament on 5 vertices. By Tm we denote a transitive tournament — an oriented graph on m vertices with all possible edges and no directed cycles.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.