Disjoint Chains Research Articles

On a long cycle in the graph of all linear extensions of a poset consisting of two disjoint chains

We prove that the only obstacle for the graph of all linear extensions of a poset, consisting of two disjoint odd chains, to have a Hamilton cycle are the two vertices of degree 1. In other words, if these two vertices are removed, the remaining graph has a Hamilton cycle. This is equivalent to the statement, that there exists a Hamilton cycle in the subgraph G( k, l⧹{0 k 1 l , 1 l 0 k } of G( k, l), where G( k, l) is the graph of all binary ( k + l)-tuples containing k zeroes and l ones.

A unidirectional ring partition problem

AbstractLet R = (V, E) be a unidirectional ring network where V corresponds to the set of nodes and E corresponds to the set of directed communication links. A partition of R divides R into several disjoint chains. For each partition P, there is associated a communication cost C(P). The optimal ring partition problem is to find a partition P* of R such that C(P*) = minpC(P). In this paper, we first formulate the ring partition problem into a recurrence relation. By solving the recurrence relation, we show that the ring partition can be accomplished distributively in one pass, i.e., the message complexity of our distributed ring partition algorithm is O(|V|), which is optimal. © 1993 by John Wiley & Sons, Inc.

Hamilton Paths in Graphs of Linear Extensions for Unions of Posets

This paper proves that if a poset Q has an even number of linear extensions and these extensions can be generated by adjacent transpositions, then linear extensions of union of poset Q and an arbitrary poset P can also be generated by adjacent transpositions. This result is then applied to posets P and Q, which are sums of disjoint chains.

Greene-Kleitman's theorem for infinite posets

It is proved that if (P⩽) is a poset with no infinite chain and k is a positive integer, then there exist a partition of P into disjoint chains C i and disjoint antichains A 1, A 2, ..., A k, such that each chain C i meets min (k, |C i|) antichains A j. We make a ‘dual’ conjecture, for which the case k=1 is: if (P⩽) is a poset with no infinite antichain, then there exist a partition of P into antichains A i and a chain C meeting all A i. This conjecture is proved when the maximal size of an antichain in P is 2.

Generating linear extensions of posets by transpositions

This paper considers the problem of listing all linear extensions of a partial order so that successive extensions differ by the transposition of a single pair of elements. A necessary condition is given for the case when the partial order is a forest. A necessary and sufficient condition is given for the case where the partial order consists of disjoint chains. Some open problems are mentioned.

On chains and Sperner k-families in ranked posets, II

A ranked poset P satisfies condition S if for all k the set of elements of the k largest ranks in P is a Sperner k-family. It satisfies condition T if for all k there exist disjoint chains in P which each meet the k largest ranks and which cover the kth largest rank. Griggs employed the theory of saturated partitions to prove that if P satisfies S it also satisfies T, and that the converse is true for posets with unimodal Whitney numbers. Here we present new proofs of these facts which do not require the existence of saturated partitions. The first result is proven with a simple network flow argument and the second is proven directly.

On chains and Sperner k-families in ranked posets

A ranked poset P has the Sperner property if the sizes of the largest rank and of the largest antichain in P are equal. A natural strengthening of the Sperner property is condition S: For all k, the set of elements of the k largest ranks in P is a Sperner k-family. P satisfies condition T if for all k there exist disjoint chains in P each of which meets the k largest ranks and which covers the kth largest rank. It is proven here that if P satisfies S, it also satisfies T, and that the converse, although in general false, is true for posets with unimodal Whitney numbers. Conditions S and T and the Sperner property are compared here with two other conditions on posets concerning the existence of certain partitions of P into chains.

On Dilworth's decomposition theorem

For every countable po set P without infinite chains there exists a partition ( C j : jϵI) of P into pairwise disjoint chains and an antichain A such that C j ∩ A≠ Ø for every iϵI.

Algebraic and Topological Properties of Connecting Networks

A connecting network is an arrangement of switches and transmission links allowing a certain set of terminals to be connected together in various combinations, usually by disjoint chains (paths): e.g., a central office, toll center, or military communications system. Some of the basic combinatory properties of connecting networks are studied in the present paper. Three of these properties are defined informally as follows: A network is rearrangeable if, given any set of calls in progress and any pair of idle terminals, the calls can be reassigned new routes (if necessary) so as to make it possible to connect the idle pair. A state of a network is a blocking state if some pair of idle terminals cannot be connected. A network is nonblocking in the wide sense if by suitably choosing routes for new calls it is possible to avoid all the blocking states and still satisfy all demands for connection as they arise, without rearranging existing calls. Finally, a network is nonblocking in the strict sense if it has no blocking states. A distance between states can be defined as the number of calls one would have to add or remove to change one state into the other. This distance defines a topology on the set of states. Also, the states can be partially ordered by inclusion in a natural way. This partial ordering and its dual define two more topologies for the set of states. The three topologies so obtained are used to characterize (i.e., give necessary and sufficient conditions for the truth of) the three properties of rearrangeability, nonblocking in the wide sense, and nonblocking in the strict sense. Each of these three properties represents a degree of abundance of nonblocking states; the mathematical concept used to express these degrees is the topological notion of denseness.