Digraphs with All Induced Directed Cycles of the Same Length are not → χ -Bounded

Abstract

For t > 2, let us call a digraph D t-chordal if all induced directed cycles in D have length equal to t. In an earlier paper, we asked for which t it is true that t-chordal graphs with bounded clique number have bounded dichromatic number. Recently, Aboulker, Bousquet, and de Verclos answered this in the negative for t = 3, that is, they gave a construction of 3-chordal digraphs with clique number at most 3 and arbitrarily large dichromatic number. In this paper, we extend their result, giving for each t > 3 a construction of t-chordal digraphs with clique number at most 3 and arbitrarily large dichromatic number, thus answering our question in the negative. On the other hand, we show that a more restricted class, digraphs with no induced directed cycle of length less than t, and no induced directed t-vertex path, have bounded dichromatic number if their clique number is bounded. We also show the following complexity result: for fixed t > 2, the problem of determining whether a digraph is t-chordal is coNP-complete.