Berge's conjecture and Aharoni-Hartman-Hoffman's conjecture for locally in-semicomplete digraphs

Let k be a positive integer and let D be a digraph. A path partition P of D is a set of vertex-disjoint paths which covers V(D). Its k-norm is defined as ∑P∈Pmin{|V(P)|,k}. A path partition is k-optimal if its k-norm is minimum among all path partitions of D. A partialk-coloring is a collection of k disjoint stable sets. A partial k-coloring C is orthogonal to a path partition P if each path P∈P meets min{|V(P)|,k} distinct sets of C. Berge (Eur J Comb 3(2):97–101, 1982) conjectured that every k-optimal path partition of D has a partial k-coloring orthogonal to it. A (path) k-pack of D is a collection of at most k vertex-disjoint paths in D. Its weight is the number of vertices it covers. A k-pack is optimal if its weight is maximum among all k-packs of D. A coloring of D is a partition of V(D) into stable sets. A k-pack P is orthogonal to a coloring C if each set C∈C meets min{|C|,k} paths of P. Aharoni et al. (Pac J Math 2(118):249–259, 1985) conjectured that every optimal k-pack of D has a coloring orthogonal to it. A digraph D is semicomplete if every pair of distinct vertices of D are adjacent. A digraph D is locally in-semicomplete if, for every vertex v∈V(D), the in-neighborhood of v induces a semicomplete digraph. Locally out-semicomplete digraphs are defined similarly. In this paper, we prove Berge’s and Aharoni–Hartman–Hoffman’s Conjectures for locally in/out-semicomplete digraphs354921931CNPQ - Conselho Nacional de Desenvolvimento Científico e TecnológicoFAPESP – Fundação de Amparo à Pesquisa Do Estado De São Paulo141216/2016-6; 311373/2015-12016/14132-4; 2015/11937-