On the Caccetta-Häggkvist Conjecture with a Forbidden Transitive Tournament

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 + 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. In the paper we are considering oriented graphs, i.e., directed graphs without loops, multiple edges and two vertices connected by edges in both directions. By Tm we denote a transitive tournament — an oriented graph on m vertices with all possible edges and no directed cycles. By ~ Cl we denote a directed cycle on l edges. We also denote the minimum outdegree as δ(G) = min v∈G deg(v). ∗The research was supported by the National Science Centre grant 2011/01/N/ST1/02341. the electronic journal of combinatorics 24(2) (2017), #P2.19 1 Whenever the graph G is known from the context we will omit it and write just δ. By blow-up of an oriented graph H we consider a graph whose vertex set is divided into v(H) equal parts with all the edges between parts placed accordingly to edges in H (between each two parts there are all edges directed in one way or no edges at all) and no edges inside the parts. We call those parts of the blow-up by blobs. By k-iterated blow-up of an oriented graph H we consider a graph which is a blow-up of (k − 1)-iterated blow-up of H, where the 1-iterated blow-up of H is just a blow-up of H. In 1978 Caccetta and Häggkvist [4] stated the following conjecture Conjecture 1 (Caccetta-Häggkvist Conjecture). Every oriented graph on n vertices without directed cycles of length less than or equal to l has δ 6 (n− 1)/l. The conjecture was proved for many large values of l by Caccetta and Häggkvist [4], Hamidoune [11], Hoáng and Reed [14], and Shen [25]. Another approach was to force a directed cycle of length at most l + c for some small c. This was proved for c = 2500 by Chvátal and Szemerédi [5], for c = 304 by Nishimura [17], and for c = 73 by Shen [26]. Small values of l are more difficult and received much attention. Even for l = 3 the conjecture is still open. We focus more on this case. Let c be the minimal constant for which each triangle-free graph has δ 6 cn. Conjectured value is c = 1/3 ≈ 0.3333. Caccetta and Häggkvist [4] proved that c 6 (3− √ 5)/2 ≈ 0.3820. Then it was improved by Bondy [3] to (2 √ 6 − 3)/5 ≈ 0.3798, Shen [24] to 3 − √ 7 ≈ 0.3542, Hamburger, Haxell, and Kostochka [10] to 0.3532, and to 0.3465 by Hladký, Král’, and Norine [13]. Recently, Sereni and Volec [27] stated even further improvement to 0.3388 using flag algebras in a sophisticated way. For more results and problems related to the Caccetta-Häggkvist Conjecture see [23]. The main obstacle, which makes this problem hard, is the fact that there are many non-isomorphic extremal examples. The same is happening for example in case of the wellknown Turán Conjecture. Bondy [3] observed that the class of extremal graphs for the Caccetta-Häggkvist Conjecture is closed under lexicographic product. Later, Razborov [21] generalized the Bondy’s construction. Here we present a new way of proving the Caccetta-Häggkvist Conjecture. The main idea is to define a set of conjectures, which lead to the Caccetta-Häggkvist Conjecture, such that for each of them the set of extremal examples is more restricted and we can use the inductive arguments in parts of the extremal graphs. This way, it will be easier to prove those partial conjectures using the graph limits methods and inductive arguments. Let us now state the main conjecture. Conjecture 2. Every oriented graph on n vertices without directed cycles of length less than or equal to l and without transitive tournament T2k+1 has

Cite this paper

@article{Grzesik2017OnTC, title={On the Caccetta-H{\"a}ggkvist Conjecture with a Forbidden Transitive Tournament}, author={Andrzej Grzesik}, journal={Electr. J. Comb.}, year={2017}, volume={24}, pages={P2.19} }