Path partitions and packs of acyclic digraphs

Abstract

Let G be an acyclic directed graph with \V(G)\>k. We prove that there exists a colouring { C x , C 2 ,..., C m } such that for every collection {P l9 P 2 ,... ,P k } of k vertex disjoint paths with |UjLi Pj\ a maximum, each colour class C, meets min{|CJ, k} of these paths. An analogous theorem, partially interchanging the roles of paths and colour classes, has been shown by Cameron [4] and Saks [17] and we indicate a third proof. Introduction.. ,ι>/) will be denoted by K(P). The cardinality of P, denoted by |P|, is |F(P)|.A family 0* of paths is called a path-partition of G if its members are vertex disjoint and U{F(P): Pe#) == V. For each nonnegative integer k 9 the k-norm \0>\ k of a path partition ^ = {P v ... y P m } is defined by A partition which minimizes 1^)^. is called k-optimum. For example, a 1-optimum partition is a partition P containing a minimum number of paths.A partial k-colouring is a family #* = {C^, C 2 ,... ,C r } of at most /: disjoint independent sets C f called colour classes. The cardinality of a partial ^-colouring #* = {C 1? C 2 ,...,C t ) is |U{ βl Q, and <g k is said to be optimum if jU^iQ is as large as possible. A path partition ^ = {P v P 2> ... ,P m } and a partial fc-colouring ^A are orthogonal if every path P. in 0> meets min{ (P^, fc} different colour classes of %> k .Berge [2] made the following conjecture:Conjecture 1. Let G be a directed graph and let A: be a positive integer. Then for every fc-optimum path partition ^, there exists a partial A>colouring #* orthogonal to 0*.

A unified approach to known and unknown cases of Berge's conjecture

“…Conjecture 2.4 holds for k = 1 by the Gallai-Milgram [11] theorem, for k ≥ by the Gallai-Roy Theorem [10,16], and for all acyclic digraphs [2,17,6]. It also holds in the case that P = P ≥k [1].…”

“…(For precise statements of all the mentioned theorems, see Section 2.) Berge's conjecture was proved for all acyclic digraphs; see [15,7,17,6,2]. For k = 1, Berge's Conjecture holds by the Gallai-Milgram Theorem [11].…”

Berge's conjecture on directed path partitions—a survey

“…Berge [10] used a different term for orthogonality, defining a k-colouring C k to be strong for a path P if C k meets P in exactly min{|P |, k} different colour classes. We prefer the concept of orthogonality as used in [33,4].…”

“…Moreover, Conjecture 2.8 was proved for acyclic digraphs in [50,21,4,57]. In fact, a stronger result holds for acyclic digraphs: Theorem 2.9 (Saks [50], Cameron [21], Aharoni et al [4]). Let G be an acyclic digraph, and let k be a positive integer.…”

“…Several researchers (see [4,21,23,29]) used LP techniques to extend the theorem of Greene and Kleitman to acyclic digraphs. We shall describe here the recent proof suggested by Pierre Charbit [25].…”