Proof of Berge’s path partition conjecture for k≥λ−3

Proof of Berge's path partition conjecture for k ≤ λ - 3

Let D be a digraph. A path partition of D is called k-optimal if the sum of the k-norms of its paths isminimal. The k-norm of a path P ismin(|V (P)|, k). Berge’s path partition conjecture claims that for every k-optimal path partition P there are k disjoint stable sets orthogonal to P . For general digraphs the conjecture has been proven for k = 1, 2, λ − 1, λ, where λ is the length of a longes...

Proof of Berge's strong path partition conjecture for k=2

Berge’s strong path partition conjecture from 1982 generalizes and extends Dilworth’s theorem and the Greene–Kleitman theorem which are well known for partially ordered sets. The conjecture is known to be true for all digraphs only for k = 1 (by the Gallai–Milgram theorem) and for k ≥ λ (where λ is the cardinality of the longest path in the graph). The attempts made, so far, to prove the conjec...

The directed path partition conjecture

The Directed Path Partition Conjecture is the following: If D is a digraph that contains no path with more than λ vertices then, for every pair (a, b) of positive integers with λ = a + b, there exists a vertex partition (A, B) of D such that no path in D〈A〉 has more than a vertices and no path in D〈B〉 has more than b vertices.We develop methods for finding the desired partitions for various cla...

The Path Partition Conjecture for Oriented Graphs

The vertex set and arc set of a digraph D are denoted by V (D) and E (D), respectively, and the number of vertices in a digraph D is denoted by n (D). A directed cycle (path, walk) in a digraph will simply be called a cycle (path, walk). A graph or digraph is called hamiltonian if it contains a cycle that visits every vertex, traceable if it contains a path that visits every vertex, and walkabl...

Proof of Andrews' Conjecture on Partition Identities