Bispindles in strongly connected digraphs with large chromatic number

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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.