Longest path partitions in generalizations of tournaments

We consider the so-called Path Partition Conjecture for digraphs which states that for every digraph, D, and every choice of positive integers, λ1, λ2, such that λ1 + λ2 equals the order of a longest directed path in D, there exists a partition of D into two digraphs, D1 and D2, such that the order of a longest path in Di is at most λi, for i = 1, 2. We prove that certain classes of digraphs, which are generalizations of tournaments, satisfy the Path Partition Conjecture and that some of the classes even satisfy the conjecture with equality. ∗Department of Mathematics and Computer Science, University of Southern Denmark, DK-5230 Odense, Denmark †Department of Mathematics and Computer Science, University of Southern Denmark, DK-5230 Odense, Denmark ‡Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, United Kingdom