Erdös-Hajnal Conjecture for New Infinite Families of Tournaments

A celebrated unresolved conjecture of Erdös and Hajnal states that for every undirected graph H there exists (H) > 0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n (H). This conjecture has a directed equivalent version stating that for every tournament H there exists (H) > 0 such that every H−free n−vertex tournament T contains a transitive subtournament of size at least n (H). Recently the conjecture was proved for all six-vertex tournaments, except K6. In this paper we construct two infinite families of tournaments for which the conjecture is still open for infinitely many tournaments in these two families − the family of so-called super nebulas and the family of so-called super triangular galaxies. We prove that for every super nebula H1 and every ∆galaxy H2 there exist (H1, H2) such that every {H1, H2}−free tournament T contains a transitive subtournament of size at least |T | (H 1 ,H 2). We also prove that for every central triangular galaxy H there exist (K6, H) such that every {K6, H}−free tournament T contains a transitive subtournament of size at least |T | (K 6 ,H). And we give an extension of our results.

2020

The celebrated Erdös-Hajnal conjecture states that for every undirected graph H there exists (H) > 0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n (H). This conjecture has a directed equivalent version stating that for every tournament H there exists (H) > 0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least n (H). This conjecture is proved for few infinite families of tournaments. In this paper we construct a new infinite family of tournaments − the family of so-called flotilla-galaxies and we prove the correctness of the conjecture for every flotilla-galaxy tournament.

A celebrated unresolved conjecture of Erdös and Hajnal states that for every undirected graph H there exists (H) > 0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n (H). The conjecture has a directed equivalent version stating that for every tournament H there exists (H) > 0 such that every H−free n−vertex tournament T contains a transitive subtournament of order at least n (H). Both the directed and the undirected versions of the conjecture are known to be true for small graphs (tournaments). So far the conjecture was proved only for some specific families of prime tournaments (Berger et al., 2014 [2]; Choromanski, 2015 [3]), tournaments constructed according to the so−called substitution procedure (Alon et al., 2001 [1]) allowing to build bigger graphs, and for all five−vertex tournaments (Berger et al. 2014 [2]). Recently the conjecture was proved for all six−vertex tournament, with one exception (Berger et al. 2018 [5]), but the question about the correctness of the conjecture for all seven−vertex tournaments remained open. In this paper we prove the correctness of the conjecture for several seven−vertex tournaments.

2022

An equivalent directed version of the celebrated unresolved conjecture of Erdös and Hajnal proposed by Alon et al. states that for every tournament H there exists ǫ(H) > 0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least n ǫ(H). A tournament H has the strong EH-property if there exists c > 0 such that for every H-free tournament T with |T | > 1, there exist disjoint vertex subsets A and B, each of cardinality at least c|T | and every vertex of A is adjacent to every vertex of B. Berger et al. proved that the unique five-vertex tournament denoted by C 5 , where every vertex has two inneighbors and two outneighbors has the strong EH-property. It is known that every tournament with the strong EH-property also has the EH-property. In this paper we construct an infinite class of tournaments − the so-called spiral galaxies and we prove that every spiral galaxy has the strong EH-property.

European Journal of Combinatorics

In this work we present a version of the so called Chen and Chvátal's conjecture for directed graphs. A line of a directed graph D is defined by an ordered pair (u, v), with u and v two distinct vertices of D, as the set of all vertices w such that u, v, w belong to a shortest directed path in D containing a shortest directed path from u to v. A line is empty if there is no directed path from u to v. Another option is that a line is the set of all vertices. The version of the Chen and Chvátal's conjecture we study states that if none of previous options hold, then the number of distinct lines in D is at least its number of vertices. Our main result is that any tournament satisfies this conjecture as well as any orientation of a complete bipartite graph of diameter three.

Discussiones Mathematicae Graph Theory, 2017

The dichromatic number dc(D) of a digraph D is defined to be the minimum number of colors such that the vertices of D can be colored in such a way that every chromatic class induces an acyclic subdigraph in D. The cyclic circulant tournament is denoted by T = − → C 2n+1 (1, 2,. .. , n), where V (T) = Z 2n+1 and for every jump j ∈ {1, 2,. .. , n} there exist the arcs (a, a + j) for every a ∈ Z 2n+1. Consider the circulant tournament − → C 2n+1 k obtained from the cyclic tournament by reversing one of its jumps, that is, − → C 2n+1 k has the same arc set as − → C 2n+1 (1, 2,. .. , n) except for j = k in which case, the arcs are (a, a − k) for every a ∈ Z 2n+1. In this paper, we prove that dc(− → C 2n+1 k) ∈ {2, 3, 4} for every k ∈ {1, 2,. .. , n}. Moreover, we classify which circulant tournaments − → C 2n+1 k are vertex-critical r-dichromatic for every k ∈ {1, 2,. .. , n} and r ∈ {2, 3, 4}. Some previous results by Neumann-Lara are generalized.

2008

A tournament is acyclically indecomposable if no acyclic autonomous set of vertices has more than one element. We identify twelve infinite acyclically indecomposable tournaments and prove that every infinite acyclically indecomposable tournament contains a subtournament isomorphic to one of these tournaments. The profile of a tournament T is the function ϕ_T which counts for each integer n the number ϕ_T(n) of tournaments induced by T on the n-element subsets of T, isomorphic tournaments being identified. As a corollary of the result above we deduce that the growth of ϕ_T is either polynomial, in which case ϕ_T(n)≃ an^k, for some positive real a, some non-negative integer k, or as fast as some exponential.

2010

Few families of tournaments satisfying the n-e.c. adjacency property are known. We supply a new random construction for generating infinite families of vertex-transitive n-e.c. tournaments by considering circulant tournaments. Switching is used to generate new n-e.c. tournaments of certain orders. With aid of a computer search, we demonstrate that there is a unique minimum order 3-e.c. tournament of order 19, and there are no 3-e.c. tournaments of orders 20, 21, and 22. We show that there are no 4-e.c. tournaments of orders 47 and 48 improving the lower bound for the minimum order of such a tournament.

Congressus numerantium, 2002

Given an acyclic digraph D, we seek a smallest sized tournament T that has D as a minimum feedback arc set. The reversing number of a digraph is defined to be r(D) = |V (T )|−|V (D)| . The case where D is a tournament Tn was studied by Isaak in 1995 using an integer linear programming formulation. In particular, this approach was used to produce lower bounds for r(Tn), and it was conjectured that the given bounds were tight. We examine the class of tournaments where n = 2 k + 2 k−2 and show the known lower bounds for r(T n ) are best possible.

Comptes Rendus Mathematique, 2010

Given a tournament T=(V,A), a subset X of $V$ is an interval of T provided that for every a, b in X and x\in V-X, (a,x) in A if and only if (b,x) in A. For example, $\emptyset$, {x}(x in V) and V are intervals of T, called trivial intervals. A tournament, all the intervals of which are trivial, is indecomposable; otherwise, it is decomposable. A critical tournament is an indecomposable tournament T of cardinality $\geq 5$ such that for any vertex x of T, the tournament T-x is decomposable. The critical tournaments are of odd cardinality and for all $n \geq 2$ there are exactly three critical tournaments on 2n+1 vertices denoted by $T_{2n+1}$, $U_{2n+1}$ and $W_{2n+1}$. The tournaments $T_{5}$, $U_{5}$ and $W_{5}$ are the unique indecomposable tournaments on 5 vertices. We say that a tournament T embeds into a tournament T' when T is isomorphic to a subtournament of T'. A diamond is a tournament on 4 vertices admitting only one interval of cardinality 3. We prove the following theorem: if a diamond and $T_{5}$ embed into an indecomposable tournament T, then $W_{5}$ and $U_{5}$ embed into T. To conclude, we prove the following: given an indecomposable tournament T, with $\mid\!V(T)\!\mid \geq 7$, T is critical if and only if the indecomposable subtournaments on 7 vertices of T are isomorphic to one and only one of the tournaments $T_{7}$, $U_{7}$ and $W_{7}$.