Bispindle in strongly connected digraphs with large chromatic number

A (k1+k2)-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 γk such that every strongly connected digraph with chromatic number greater than γk contains a (2+1)-bispindle with the (y, x)-dipath and one of the (x, y)-dipaths of length at least k.