On reorienting graphs by pushing down maximal vertices

We study the operation of pushing down elements in the diagram of a finite ordered set. Two natural questions about this operation are, ‘Which orientations of the underlying graph can be obtained from a given orientation by pushing down?’ and ‘Which sets of vertices can become the sets of maximal elements in such orientations?’. For both questions thére are easy necessary conditions. We show that these conditions are also sufficient. The results are extended to cover all induced subgraphs and arbitrary orientations of a finite graph.     

Planar acyclic oriented graphs

A plane Hasse representation of an acyclic oriented graph is a drawing of the graph in the Euclidean plane such that all arcs are straight-line segments directed upwards and such that no two arcs cross. We characterize completely those oriented graphs which have a plane Hasse representation such that all faces are bounded by convex polygons. From this we derive the Hasse representation analogue, due to Kelly and Rival of Fary's theorem on straight-line representations of planar graphs and the Kuratowski type theorem of Platt for acyclic oriented graphs with only one source and one sink. Finally, we describe completely those acyclic oriented graphs which have a vertex dominating all other vertices and which have no plane Hasse representation, a problem posed by Trotter.     

Minimizing the jump number for partially ordered sets: A graph-theoretic approach

The purpose of this paper is to present a graph-theoretic approach to the jump number problem for N-free posets which is based on the observation that the Hasse diagram of an N-free poset is a line digraph. Therefore, to every N-free poset P we can assign another digraph which is the root digraph of the Hasse diagram of P. Using this representation we show that the jump number of an N-free poset is equal to the cyclomatic number of its root digraph and can be found (without producing any linear extension) by an algorithm which tests if a given poset is N-free. Moreover, we demonstrate that there exists a correspondence between optimal linear extensions of an N-free poset and spanning branchings of its root digraph. We provide also another proof of the fact that optimal linear extensions of N-free posets are exactly greedy linear extensions. In conclusion, we discuss some possible generalizations of these results to arbitrary posets.     

Path-closed sets

Given a digraphG = (V, E), call a node setT ⊆V path-closed ifv, v′ εT andw εV is on a path fromv tov′ impliesw εT. IfG is the comparability graph of a posetP, the path-closed sets ofG are the convex sets ofP. We characterize the convex hull of (the incidence vectors of) all path-closed sets ofG and its antiblocking polyhedron inR ^v , using lattice polyhedra, and give a minmax theorem on partitioning a given subset ofV into path-closed sets. We then derive good algorithms for the linear programs associated to the convex hull, solving the problem of finding a path-closed set of maximum weight sum, and prove another min-max result closely resembling Dilworth’s theorem.     

Inequalities for the greedy dimensions of ordered sets

Every linear extension L: [x ₁<x ₂<...<x _m ] of an ordered set P on m points arises from the simple algorithm: For each i with 0i<m, choose x _i+1 as a minimal element of P–{x _j :ji}. A linear extension is said to be greedy, if we also require that x _i+1 covers x _i in P whenever possible. The greedy dimension of an ordered set is defined as the minimum number of greedy linear extensions of P whose intersection is P. In this paper, we develop several inequalities bounding the greedy dimension of P as a function of other parameters of P. We show that the greedy dimension of P does not exceed the width of P. If A is an antichain in P and |P–A|2, we show that the greedy dimension of P does not exceed |P–A|. As a consequence, the greedy dimension of P does not exceed |P|/2 when |P|4. If the width of P–A is n and n2, we show that the greedy dimension of P does not exceed n ²+n. If A is the set of minimal elements of P, then this inequality can be strengthened to 2n–1. If A is the set of maximal elements, then the inequality can be further strengthened to n+1. Examples are presented to show that each of these inequalities is best possible.Research supported in part by the National Science Foundation under ISP-80110451.Research supported in part by the National Science Foundation under ISP-80110451 and DMS-8401281.     

A theory of recursive dimension for ordered sets

The classical theorem of R. P. Dilworth asserts that a partially ordered set of width n can be partitioned into n chains. Dilworth's theorem plays a central role in the dimension theory of partially ordered sets since chain partitions can be used to provide embeddings of partially ordered sets in the Cartesian product of chains. In particular, the dimension of a partially-ordered set never exceeds its width. In this paper, we consider analogous problems in the setting of recursive combinatorics where it is required that the partially ordered set and any associated partition or embedding be described by recursive functions. We establish several theorems providing upper bounds on the recursive dimension of a partially ordered set in terms of its width. The proofs are highly combinatorial in nature and involve a detailed analysis of a 2-person game in which one person builds a partially ordered set one point at a time and the other builds the partition or embedding.This paper was prepared while the authors were supported, in part, by NSF grant ISP-80-11451. In addition, the second author received support under NSF grant MCS-80-01778 and the third author received support under NSF grant MCS-82-02172.     

On minimizing jumps for ordered sets

An ordered set P is called K-free if it does not contain a four-element subset {a, b, c, d} such that a < b is the only comparability among these elements. In this paper we present a polynomial algorithm to find the jump number of K-free ordered sets.     

Decompositions of partially ordered sets into chains and antichains of given size

Every partially ordered set P on at least (1+o(1))n ³ elements can be decomposed into subposets of size n that are almost chains or antichains. This lower bound on P is asymptotically best possible. Similar results are presented for other types of combinatorial structures.Research supported in part by the AKA Research Fund of the Hungarian Academy of Sciences, grant 1-3-86-264.     

Chain partitions of ordered sets

For natural numbers s and r and a finite poset P of height h, there exists a finite poset P of height H(s, r, h) such that for an arbitrary r-colouring of the s-chains of P, a monochromatically embedded copy of P can be found in P. A best possible upper bound for H(s, r, h) in terms of the well-known Ramsey numbers is given.     

The chromatic number of the product of two ℵ1-chromatic graphs can be countable

We prove (in ZFC) that for every infinite cardinal ϰ there are two graphsG ₀,G ₁ with χ(G ₀)=χ(G ₁)=ϰ⁺ and χ(G ₀×G ₁)=ϰ. We also prove a result from the other direction. If χ(G ₀)≧≧ℵ₀ and χ(G ₁)=k<ω, then χ(G ₀×G ₁)=k.     

On the size of jump-critical ordered sets

The maximum size of a jump-critical ordered set with jump-number m is at most (m+1)!     

Chromatic number of Hasse diagrams,eyebrows and dimension

We construct posets of dimension 2 with highly chromatic Hasse diagrams. This solves a previous problem by Nesetril and Trotter.     

On the dimensions of ordered sets of bounded degree

Let P be a partially ordered set. Define k = k (P) = max _p |{x P : p < x or p = x}|, i.e., every element is comparable with at most k others. Here it is proven that there exists a constant c (c < 50) such that dim P < ck(log k)². This improves an earlier result of Rödl and Trotter (dim P 2 k ²+2). Our proof is nonconstructive, depending in part on Lovász' local lemma.Supported in part by NSF under Grant No. MCS83-01867 and by a Sloan Research Fellowship.     

3-Interval irreducible partially ordered sets

In this paper we discuss the characterization problem for posets of interval dimension at most 2. We compile the minimal list of forbidden posets for interval dimension 2. Members of this list are called 3-interval irreducible posets. The problem is related to a series of characterization problems which have been solved earlier. These are: The characterization of planar lattices, due to Kelly and Rival [5], the characterization of posets of dimension at most 2 (3-irreducible posets) which has been obtained independently by Trotter and Moore [8] and by Kelly [4] and the characterization of bipartite 3-interval irreducible posets due to Trotter [9].We show that every 3-interval irreducible poset is a reduced partial stack of some bipartite 3-interval irreducible poset. Moreover, we succeed in classifying the 3-interval irreducible partial stacks of most of the bipartite 3-interval irreducible posets. Our arguments depend on a transformationP B(P), such that IdimP=dimB(P). This transformation has been introduced in [2].Supported by the DFG under grant FE 340/2–1.     

Random graphs and covering graphs of posets

For a graph G, we define c(G) to be the minimal number of edges we must delete in order to make G into a covering graph of some poset. We prove that, if p=n ^-1+(n) ,where (n) is bounded away from 0, then there is a constant k ₀>0 such that, for a.e. G _p , c(G _p )k ₀ n ¹⁺⁽ⁿ⁾ .In other words, to make G _p into a covering graph, we must almost surely delete a positive constant proportion of the edges. On the other hand, if p=n ^-1+(n) , where (n)0, thenc(G _p )=o(n ¹⁺⁽ⁿ⁾ ), almost surely.Partially supported by MCS Grant 8104854.     

Powers of ordered sets

Recently there has been significant progress in the study of powers of ordered sets. Much of this work has concerned cancellation laws for powers and uses these two steps. First, logarithmic operators are introduced to transform cancellation problems for powers into questions involving direct product decompositions. Second, refinement theorems for direct product decompositions are brought to bear. Here we present two results with the aim of highlighting these steps.Supported by NSF grant MCS 83-02054     

Minimizing setups in ordered sets of fixed width

A simply polynomial time algorithm is given for computing the setup number, or jump number, of an ordered set with fixed width. This arises as an interesting application of a polynomial time algorithm for solving a more general weighted problem in precedence constrained scheduling.     

Ladders in ordered sets

An ordered set P is said to have the 2-cutset property if for every element x of P there is a subset S of P whose elements are noncomparable to x, such that |S|2 and such that every maximal chain in P meets {x}S. It is shown that if P has the 2-cutset property and has width n then P contains a ladder of length [1/2(n–3)].     

On oriented graphs whose skew spectral radii do not exceed 2

Let S(G^σ)$S(G^{\sigma })$ be the skew adjacency matrix of the oriented graph G^σ$G^{\sigma }$ of order n   and _λ1,_λ2,…,_λn${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be all eigenvalues of S(G^σ)$S(G^{\sigma })$. The skew spectral radius _ρs(G^σ)${\rho }_s(G^{\sigma })$ of G^σ$G^{\sigma }$ is defined as max{|_λ1|,|_λ2|,…,|_λn|}$\max \{|{\lambda }_1|,|{\lambda }_2|,\dots ,|{\lambda }_n|\}$. In this paper, we investigate oriented graphs whose skew spectral radii do not exceed 2.