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We consider a special packing-covering pair of problems. The
packing problem is a natural generalization of finding a
(weighted) maximum independent set in an interval graph, the
covering problem generalizes the problem of finding a (weighted)
minimum clique cover in an interval graph. The problem pair
involves weights and capacities; we consider the case of unit
weights and the case of unit capacities. In each case we describe
a simple algorithm that outputs a solution to the packing problem
and...
An additive induced-hereditary property of graphs is any class of finite simple graphs which is closed under isomorphisms, disjoint unions and induced subgraphs. The set of all additive induced-hereditary properties of graphs, partially ordered by set inclusion, forms a completely distributive lattice. We introduce the notion of the join-decomposability number of a property and then we prove that the prime ideals of the lattice of all additive induced-hereditary properties are divided into two groups,...
We describe the class of probability measures whose moments are given in terms of the Aval numbers. They are expressed as the multiplicative free convolution of measures corresponding to the ballot numbers .
Let G be a finite group of even order. We give some bounds for the probability p(G) that a randomly chosen element in G has a square root. In particular, we prove that p(G) ≤ 1 - ⌊√|G|⌋/|G|. Moreover, we show that if the Sylow 2-subgroup of G is not a proper normal elementary abelian subgroup of G, then p(G) ≤ 1 - 1/√|G|. Both of these bounds are best possible upper bounds for p(G), depending only on the order of G.
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