On Linial's Conjecture For Spine Digraphs

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)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 Sigma(p is an element of p) min{vertical bar V(P)vertical bar, k}. Let pi(k)(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 pi(k)(D). We prove this conjecture for a class of digraphs we call spine digraphs, which is a proper generalization of split digraphs. (C) 2017 Elsevier B.V. All rights reserved.3405851854National Counsel of Technological and Scientific Development of Brazil (CNPq) [141216/2016-6, 311373/2015-1, 477692/2012-5]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq