Linial's Conjecture for Arc-spine Digraphs

A path partition $P$ of a digraph D is a collection of directed paths such that every vertex belongs to precisely one path. Given a positive integer k, the k-norm of a path partition $P$ of D is defined as ${\sum }_{P\in P}\min \left\{\right|P_i|,k\}$. A path partition of a minimum k-norm is called k-optimal and its k-norm is denoted by π_k(D). A stable set of a digraph D is a subset of pairwise non-adjacent vertices of V(D). Given a positive integer k, we denote by α_k(D) the largest set of vertices of D that can be decomposed into k disjoint stable sets of D. In 1981, Linial conjectured that π_k(D) ≤ α_k(D) for every digraph. We say that a digraph D is arc-spine if V(D) can be partitioned into two sets X and Y where X is traceable and Y contains at most one arc in A(D). In this paper we show the validity of Linial's Conjecture for arc-spine digraphs.