Subdivisions of oriented cycles in Hamiltonian digraphs with small chromatic number

Abstract

Cohen et al. conjectured that, for each oriented cycle C, there exists an integer f(C) such that any f(C)-chromatic strong digraph contains a subdivision of C as a subdigraph. In the same paper, Cohen et al. proved this conjecture for cycles with two blocks by showing that the chromatic number of strong digraphs that include no subdivision of a cycle with two blocks, with lengths of k1 and k2, is bounded from above by O((k1+k2)4). More recently, Kim et al. improved this upper bound to O((k1+k2)2) for the class of strong digraphs and to O(k1+k2) for the class of Hamiltonian digraphs. In this paper, we confirm Cohen et al.'s conjecture for Hamiltonian digraphs. We demonstrate a stronger version by showing that every 3n-chromatic Hamiltonian digraph contains a subdivision for every oriented cycle of order n. © 2022 Elsevier B.V.