57 research outputs found
Extending the Greene-Kleitman theorem to directed graphs
AbstractThe celebrated Dilworth theorem (Ann. of Math. 51 (1950), 161–166) on the decomposition of finite posets was extended by Greene and Kleitman (J. Combin. Theory Ser. A 20 (1976), 41–68). Using the Gallai-Milgram theorem (Acta Sci. Math. 21 (1960), 181–186) we prove a theorem on acyclic digraphs which contains the Greene-Kleitman theorem. The method of proof is derived from M. Saks' elegant proof (Adv. in Math. 33 (1979), 207–211) of the Greene-Kleitman theorem
Berge's conjecture on directed path partitions—a survey
AbstractBerge'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>1.11Only recently it was proved Berger and Ben-Arroyo Hartman [56] for k=2 (added in proof). In this paper, we will survey partial results on the conjecture, look into different proof techniques for these results, and relate the conjecture to other theorems, conjectures and open problems of Berge and other mathematicians
On the central levels problem
The \emph{central levels problem} asserts that the subgraph of the -dimensional hypercube induced by all bitstrings with at least many 1s and at most many 1s, i.e., the vertices in the middle levels, has a Hamilton cycle for any and .
This problem was raised independently by Buck and Wiedemann, Savage, Gregor and {\v{S}}krekovski, and by Shen and Williams, and it is a common generalization of the well-known \emph{middle levels problem}, namely the case , and classical binary Gray codes, namely the case .
In this paper we present a general constructive solution of the central levels problem.
Our results also imply the existence of optimal cycles through any sequence of consecutive levels in the -dimensional hypercube for any and .
Moreover, extending an earlier construction by Streib and Trotter, we construct a Hamilton cycle through the -dimensional hypercube, , that contains the symmetric chain decomposition constructed by Greene and Kleitman in the 1970s, and we provide a loopless algorithm for computing the corresponding Gray code
Linial’s Conjecture for Arc-spine Digraphs
A path partition P of a digraph D is a collection of directed paths such that every vertex belongs to precisely one path. Given a positive integer k, the k-norm of a path partition P of D is defined as Sum (p in P) min{|p_i|, k}. A path partition of a minimum k-norm is called k-optimal and its k-norm is denoted by π_k(D). A stable set of a digraph D is a subset of pairwise non-adjacentvertices of V(D). Given a positive integer k, we denote by alpha_k(D) the largest set of vertices of D that can be decomposed into k disjoint stable sets of D. In 1981, Linial conjectured that pi_k(D) ≤ alpha_k(D) for every digraph. We say that a digraph D is arc-spine if V(D) can be partitioned into two sets X and Y where X is traceable and Y contains at most one arc in A(D). In this paper we show the validity of Linial’s Conjecture for arc-spine digraphs
A class of matroids derived from saturated chain partitions of partially ordered sets
AbstractThe simultaneously k- and (k − 1)-saturated chain partitions of a finite partially ordered set P determine a matroid Gk(P). This matroid is a gammoid. The identity on P induces a strong map from Gk(P) to Gk + 1(P). This strong map has a linear representation
On Path Partitions and Colourings in Digraphs
Abstract. We provide a new proof of a theorem of Saks which is an extension of Greene's Theorem to acyclic digraphs, by reducing it to a similar, known extension of Greene and Kleitman's Theorem. This suggests that the Greene-Kleitman Theorem is stronger than Greene's Theorem on posets. We leave it as an open question whether the same holds for all digraphs, that is, does Berge's conjecture concerning path partitions in digraphs imply the extension of Greene's theorem to all digraphs (conjecture
On the heapability of finite partial orders
We investigate the partitioning of partial orders into a minimal number of
heapable subsets. We prove a characterization result reminiscent of the proof
of Dilworth's theorem, which yields as a byproduct a flow-based algorithm for
computing such a minimal decomposition. On the other hand, in the particular
case of sets and sequences of intervals we prove that this minimal
decomposition can be computed by a simple greedy-type algorithm. The paper ends
with a couple of open problems related to the analog of the Ulam-Hammersley
problem for decompositions of sets and sequences of random intervals into
heapable sets
- …