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 (1, 1)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 strongly connected digraphs with large chromatic number contains a (2, 1)-bispindle, where at least one of the (x, y)-dipaths and the (y, x)-dipath are long.