Cyclic Triangle Factors in Regular Tournaments

Lina Li,Theodore Molla

doi:10.37236/7759

Abstract

Both Cuckler and Yuster independently conjectured that when $n$ is an odd positive multiple of $3$ every regular tournament on $n$ vertices contains a collection of $n/3$ vertex-disjoint copies of the cyclic triangle. Soon after, Keevash \& Sudakov proved that if $G$ is an orientation of a graph on $n$ vertices in which every vertex has both indegree and outdegree at least $(1/2 - o(1))n$, then there exists a collection of vertex-disjoint cyclic triangles that covers all but at most $3$ vertices. In this paper, we resolve the conjecture of Cuckler and Yuster for sufficiently large $n$.

Highlights

Let H and G be graphs or directed graphs
The celebrated Hajnal-Szemeredi Theorem [7] states that for every positive integer r, if n is a positive multiple of r and G is a graph on n vertices such that δ(G) (1 − 1/r)n, G contains a Kr-factor
We use a very small part of the theory and notation developed in [11], [10],[8],[15], and [17]. It is based on the absorbing method of Rodl, Rucinski and Szemeredi [20]. This theory was developed for hypergraphs, so we define, for every oriented graph G, the hypergraph H(G) to be the 3-uniform hypergraph in which xyz is in an edge if and only if xyz is a cyclic triangle in G

Summary

Introduction

Let H and G be graphs or directed graphs. An H-tiling of G is a collection of vertexdisjoint copies of H in G. What is the smallest φ 0 such that there exists n0 such that for every n n0 every oriented graph G on n vertices with δ0(G) (4/9 + φ)n contains either a divisibility barrier or a cyclic triangle factor?. For some ψ > 1/3 and all sufficiently large n, there exists a C3-free oriented graph on n vertices with minimum semidegree ψn, a similar example would imply that φ must be strictly greater than 0 in Problem 6. Note that such a C3-free oriented graph would imply that the famous Caccetta-Haggkvist Conjecture [2] is false.

Additional Definitions and Notation

Overview

Proofs of Lemmas 11 and 12

Proof of Lemma 13

Proof of Lemma 14

Proof of Theorem 7 - The Extremal Case