Subdivisions of oriented cycles in digraphs with large chromatic number

Abstract

An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle C, there are digraphs containing no subdivision of C (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any C a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of C. We prove a similar result for the antidirected cycle on four vertices (in which two vertices have out‐degree 2 and two vertices have in‐degree 2).

Cycles with two blocks in k‐chromatic digraphs

“…In this article, we improve the above upper bound (( + ) 4 ) to (( + ) 2 ) using a quite different approach from [6]. The following is our main result.…”

“…We mention that a different extension of Bondy's theorem was obtained in [5]. Very recently, among other results, Cohen, Havet, Lochet, and Nisse [6] first obtained a finite upper bound of ( ) for strong digraphs containing no ( , ). Precisely, they proved the following.…”

“…■ It will be interesting to improve the upper bound of Theorem 1.4 further, for instance, to ( + ). We direct interested readers to [6] and the survey [12] for many related problems.…”

“…Precisely, they proved the following. Theorem 1.3 (Cohen, Havet, Lochet, and Nisse [6]). Let and be integers such that ≥ ≥ 2 and ≥ 4, and a strong digraph with no 2-block cycle ( , ).…”

On the Complexity of Finding Internally Vertex-Disjoint Long Directed Paths

“…The existence of subdivisions of 2-spindles has attracted some interest in the literature. Indeed, Benhocine and Wojda [4] showed that a tournament on n ≥ 7 vertices always contains a subdivision of a ( 1 , 2 )-spindle such that 1 + 2 = n. More recently, Cohen et al [9] showed that a strongly connected digraph with chromatic number Ω(( 1 + 2 ) 4 ) contains a subdivision of a ( 1 , 2 )-spindle, and this bound was subsequently improved to Ω(( 1 + 2 ) 2 ) by Kim et al [20], who also provided improved bounds for Hamiltonian digraphs.We consider two problems concerning the existence of subdivisions of 2-spindles. The first one is, given an input digraph G, find the largest integer such that G contains a subdivision of a ( 1 , 2 )-spindle with min{ 1 , 2 } ≥ 1 and 1 + 2 = .…”