Bispindle in strongly connected digraphs with large chromatic number

A ( k 1 + k 2 )-bispindle is the union of k1 (x, y)-dipaths and k2 (y, x)-dipaths, all these dipaths being pairwise internally disjoint. Recently, Cohen et al. showed that for every (2 + 0)-bispindle B, there exists an integer k such that every strongly connected digraph with chromatic number greater than k contains a subdivision of B. We investigate generalisations of this result by first showing constructions of strongly connected digraphs with large chromatic number without any ( 3 + 0 )-bispindle or ( 2 + 2 )-bispindle. Then we show that for any k, there exists γ k such that every strongly connected digraph with chromatic number greater than γ k contains a ( 2 + 1 )-bispindle with the (y, x)-dipath and one of the (x, y)-dipaths of length at least k.