On Linial's conjecture for spine digraphs

A path partition P of a digraph D is a set of disjoint paths which covers V(D). Let k be a positive integer. The k-norm of a path partition P of a digraph is defined as ∑P∈Pmin{|V(P)|,k}. Let πk(D) denote the smallest k-norm among all path partitions of a digraph D. In 1981, Linial conjectured that for any positive integer k each digraph D contains k disjoint stable sets whose union has size at least πk(D). We prove this conjecture for a class of digraphs we call spine digraphs, which is a proper generalization of split digraphs.