A graph is h-perfect if its stable set polytope can be completely described by nonnegativity, clique and odd-hole constraints. It is t-perfect if it furthermore has no clique of size 4. For every graph G and every c ∈ Z (G) + , the weighted chromatic number of (G, c) is the minimum cardinality of a multi-set F of stable sets of G such that every v ∈ V (G) belongs to at least cv members of F . W...

All Lie algebras and representations will be assumed to be finite dimensional over the complex numbers. Let V (m) be the irreducible sl(2)module with highest weight m ≥ 1 and consider the perfect Lie algebra g = sl(2) n V (m). Recall that a g-module is uniserial when its submodules form a chain. In this paper we classify all uniserial g-modules. The main family of uniserial g-modules is actuall...

In 1982, Padberg and Rao gave a polynomial-time separation algorithm for b-matching polyhedra. The current best known implementations of their separation algorithm run inO(|V ||E| log(|V |/|E|)) time when there are no edge capacities, but in O(|V ||E| log(|V |/|E|)) time when capacities are present. We propose a new exact separation algorithm for the capacitated case which has the same running ...

We present two ways of adjoining a perfect set of mutually random reals to a model V of ZFC. We also investigate the existence of perfect free subsets for projective functions f : (ω) → ω. 1991 Mathematics subject classification. 03E05 03E15 03E35 03E40

Given 1 1n € Z, the n-cube Q, is the graph with vertex set V : {0, l}' and ed"geset n:VT:orfi, where f' : {(r,w) e V xV: (u _ w)i:1if and only if j: i,(j:0,.'. ,n-l)), for i:0,... ,n-1. The edges of/t aresaidtohave (ortolie along) the directioni,for i 0,.. . ,n-!.Given a graph G, we say that S gVG) is a perfect dom'inating sei (or PDS) of G if every u € V(G) \ S is neighbor of exactly one verte...

Today, we will use an algebraic approach to solve the matching problem. Our goal is to derive an algebraic test for deciding if a graph G = (V, E) has a perfect matching. We may assume that the number of vertices is even since this is a necessary condition for having a perfect matching. First, we will define a few basic needed notations.

A matching in a graph G is a set M = {e1, e2, . . . , ek} of edges such that each vertex v ∈ V (G) appears in at most one edge of M . That is, ei ∩ ej = ∅ for all i, j. The size of a matching is the number of edges that appear in the matching. A perfect matching in a graph G is a matching in which every vertex of G appears exactly once, that is, a matching of size exactly n/2. Note that a perfe...

We study the weight distribution of a perfect coloring (equitable partition) of a graph with respect to a completely regular code (in particular, with respect to a vertex if the graph is distance-regular). We show how to compute this distribution by the knowledge of the percentage of the colors over the code. For some partial cases of completely regular codes we derive explicit formulas of weig...