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Abstract: Berge's conjecture from 1982 on path partitions in directed graphs generalizes and extends Dilworth's Theorem and the Greene-Kleitman Theorem which are well known for partially ordered sets. The conjecture relates path partitions to a collection of k independent sets, for each k ≥ 1. The conjecture is still open and intriguing for all k > 1. In this paper, we will survey partial results on the conjecture, look into different proof techniques for these results, and relate the conjecture to... (Update)
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BibTeX entry: (Update)
@misc{ hartman-berges,
author = "Irith Ben-Arroyo Hartman",
title = "{Berge's} Conjecture on Directed Path Partitions --- A Survey",
url = "citeseer.ist.psu.edu/ben-arroyohartman04berges.html" }
Citations (may not include all citations):537 Graph Theory with Applications (context) - Bondy, Murty - 1972
108 A decomposition theorem for partially ordered sets (context) - Dilworth - 1950
83 Princeton Univ (context) - Ford, Fulkerson et al. - 1962
13 The Strong Perfect Graph Theorem (context) - Chudnovsky, Robertson et al.
13 The structure of Sperner k-families (context) - Greene, Kleitman - 1976
12 Some partitions associated with a partially ordered set (context) - Greene - 1976
11 On chain and antichain families of a partially ordered set (context) - Frank - 1980
10 Orthogonal structures in directed graphs - Felsner - 1993
10 Strongly perfect graphs (context) - Berge, Duchet - 1984
7 On directed paths and circuits (context) - Gallai - 1968
7 On partitions of a partially ordered set (context) - Ho, Schwartz - 1977
6 A short proof of the Chen-Manalastas theorem (context) - Bondy - 1995
5 Verallgemeinerung eines graphentheoretischen satzes von Rede.. (context) - Gallai, Milgram - 1960
5 A Chvatal-Erdos condition (context) - Jackson, Ordaz
4 Path partitions and packs of acyclic digraphs (context) - Aharoni, Hartman et al. - 1985
4 Polyhedral and Algorithmic Ramifications of Antichains (context) - Cameron - 1982
4 Every finite strongly connected digraph of stability two has.. (context) - Chen - 1983
3 optimal partitions of a directed graph (context) - Berge - 1982
3 Diconnected orientations and a conjecture of Las Vergnas (context) - Bondy - 1976
2 On Gallai-type min-max inequalities (context) - Cameron
2 On Greene-Kleitman's theorem for general digraphs Discrete M.. (context) - Aharoni, Hartman - 1993
2 Path partitions in directed graphs and posets (context) - Berge
2 Progress in Graph Theory (context) - Berge, of et al. - 1984
2 optimum dipath partitions and partial k-colourings of acycli.. (context) - Cameron, k- - 1986
1 Variations on the Gallai-Milgram theorem (context) - Hartman - 1988
1 A note concerning paths and independence number in digraphs (context) - Hahn, Jackson - 1990
1 An Extension of the Greene and Greene-Kleitman Theorems to a.. - Hartman - 2002
1 Coflow polyhedra (context) - Cameron, Edmonds - 1992
1 On Greene's theorem for general digraphs (context) - Hartman, Salleh et al. - 1994
1 Sur un conjecture relative au probleme des codes optimaux (context) - Berge - 1962
1 Chemins et circuits des graphs complets (context) - Camion - 1959
1 Spanning a strong digraph by # circuits: A proof of Gallai's.. (context) - Bessy, Thomasse
1 Extending Greene's theorem to directed graphs (context) - Ho - 1983
1 personnal commumnication (context) - Charbit - 2004
1 A Chvatal-Erdos condition for Hamilton cycles in digraphs (context) - Jackson - 1978
1 A note on Hamilton circuits (context) - Chvatal, Erdos - 1977
1 Contributions to linear inequalities and related topics (context) - Dantzig, Ho et al.
1 optimal path partitions of acyclic digraphs (context) - Aharoni, Hartman et al. - 1983
1 An alternating path approach to path partitions and packs (context) - Aharoni, Hartman - 1985
1 Diperfect graphs (context) - Berge - 1982
1 Optimal Partitions and Packings of Paths in Directed Graphs (context) - Hartman - 1986
1 On matroid intersection (context) - Groflin, Ho - 1981
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