Soukaina Zayat

A cycle C(k1,k2,...,kn) is the oriented cycle formed of n blocks of lengths k1,k2,...,kn-1 and kn respectively. In 2018 Cohen et al. conjectured that for every positive integers k1,k2,...,kn there exists a constant g(k1,k2,...,kn) such that every strongly connected digraph containing no subdivisions of C(k1,k2,...,kn) has a chromatic number at most g(k1,k2,...,kn). In their paper, Cohen et al. confirmed the conjecture for cycles with two blocks and for cycles with four blocks having all its blocks of length 1. Recently, the conjecture was proved for special types of four-blocks cycles. In this paper, we confirm Cohen et al.'s conjecture for all six-blocks cycles C(k,1,1,1,1,1). Precisely, for any integer k, we prove that every strongly connected digraph containing no subdivisions of C(k,1,1,1,1,1) has a chromatic number at most O(k), and we significantly reduce the chromatic number in case k=1.

An equivalent directed version of the celebrated unresolved conjecture of Erdos and Hajnal proposed by Alon et al. states that for every tournament H there exists epsilon(H) > 0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least n^(epsilon(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 C5, 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.

Erdös–Hajnal conjecture states that for every undirected graph there exists such that every undirected graph on vertices that does not contain as an induced subgraph contains a clique or a stable set of size at least . This conjecture has a directed equivalent version stating that for every tournament there exists such that every ‐free ‐vertex tournament contains a transitive subtournament of order at least . This conjecture is known to hold for a few infinite families of tournaments. In this article we construct two new infinite families of tournaments—the family of so‐called galaxies with spiders and the family of so‐called asterisms, and we prove the correctness of the conjecture for these two families.