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

Not all phylogenetic networks are leaf-reconstructible

Unrooted phylogenetic networks are graphs used to represent reticulate evolutionary relationships. Accura

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

On the orders of directly indecomposable groups

AbstractWe investigate the set of those integers n for which directly indecomposable groups of order n exist. For even n such groups are easily constructed. In contrast, we show that the density of the set of odd numbers with this property is zero. For each n we define a graph whose connected components describe uniform direct decompositions of all groups of order n. We prove that for almost all odd numbers (i.e., with the exception of a set of density zero) this graph has a single ‘big’ connected component and all other vertices are isolated. We also give an asymptotic formula for the number of isolated vertices of the graph, i.e., for the number of prime divisors q of n such that every group of order n has a cyclic direct factor of order q