Evolutionary trees: an integer multicommodity max-flow-min-cut theorem

In biomathematics, the extensions of a leaf-colouration of a binary tree to the whole vertex set with minimum number of colour-changing edges are extensively studied. Our paper generalizes the problem for trees; algorithms and a Menger-type theorem are presented. The LP dual of the problem is a multicommodity flow problem, for which a max-flow-min-cut theorem holds. The problem that we solve is an instance of the NP-hard multiway cut problem

Towards random uniform sampling of bipartite graphs with given degree sequence

In this paper we consider a simple Markov chain for bipartite graphs with given degree sequence on $n$ vertices. We show that the mixing time of this Markov chain is bounded above by a polynomial in $n$ in case of {\em semi-regular} degree sequence. The novelty of our approach lays in the construction of the canonical paths in Sinclair's method.Comment: 47 pages, submitted for publication. In this version we explain explicitly our main contribution and corrected a serious flaw in the cycle decompositio

Splitting Property in Infinite Posets

It is known that in every finite poset P any maximal antichain S with some denseness property may be partitioned into disjoint subsets S 1 and S 2 , such that the union of the downset of S 1 with the upset of S 2 yields the entire poset: D(S 1 ) [U(S 2 ) = P. Hereby we give analogues results for infinite posets

Pseudo-LYM inequalities and AZ identities

We give pseudo-LYM inequalities in some posets and give a new restriction in this way for their antichains. Typically these posets fail the LYM inequality and some of them are known not to be Sperner. In the latter class our inequality yields an upper bound for the maximum size of an antichain. 1 Introduction Let us be given a ranked partially ordered set P, in which the set of elements of rank r are denoted by P r . (Following tradition and natural notation, the smallest rank in a poset is either 0 or 1.) The profile of an antichain A in P is the sequence of cardinalities f r = jA " P r j. The poset P satisfies the LYM inequality ([6], [12], [13] and [17]) if for every antichain A X r f r jP r j 1: A vast amount of literature investigates which posets have the LYM property. The LYM property has become a central issue due to two facts: (a) the LYM property seems to be the most straightforward mean to prove that a poset has the Sperner property, i.e. the largest antichain has s..

All Maximum 2-part Sperner Families

AbstractLet X = X1 ∪ X2, X1 ∩ X2 = 0 be a partition of an n-element set. Suppose that the family F of some subsets of X satisfy the following condition: if there is an inclusion F1 ⊈ F2 (F1, F2 ϵ F) in F, the difference F2 − F1 cannot be a subset of X1 or X2. Kleitman (Math. Z. 90 (1965), 251–259) and Katona (Studia Sci. Math. Hungar. 1 (1966), 59–63) proved 20 years ago that |F| is at most n choose n2. We determine all families giving equality in this theorem

Convex hulls of more-part Sperner families

The convex hulls of more-part Sperner families is defined and studied. Corollaries of the results are some well-known theorems on 2 or 3-part Sperner families. Some methods are presented giving new theorems