No abstract

We introduce a family of vertex-transitive graphs with specified subgroups of automorphisms which generalise Kneser graphs, powers of complete graphs and Cayley graphs of permutations. We compute the stability ratio for a wide class of these. Under certain conditions we characterise their stable sets of maximal size. (C) 2003 Elsevier Ltd. All rights reserved

Abstract. We call two permutations of the first n naturals colliding if they map at least one number to consecutive naturals. We give bounds for the exponential asymptotics of the largest cardinality of any set of pairwise colliding permutations of [n]. We relate this problem to the determination of the Shannon capacity of an infinite graph and initiate the study of analogous problems for infinite graphs with finite chromatic number.

Let D be the set of isomorphism types of finite double partially ordered sets, that is sets endowed with two partial orders. On ZD we define a product and a coproduct, together with an internal product, that is, degree-preserving. With these operations ZD is a Hopf algebra. We define a symmetric bilinear form on this Hopf algebra: it counts the number of pictures (in the sense of Zelevinsky) between two double posets. This form is a Hopf pairing, which means that product and coproduct are adjoint each to another. The product and coproduct correspond respectively to disjoint union of posets and to a natural decomposition of a poset into order ideals. Restricting to special double posets (meaning that the second order is total), we obtain a notion equivalent to Stanley's labelled posets, and a Hopf subalgebra already considered by Blessenohl and Schocker. The mapping which maps each double poset onto the sum of the linear extensions of its first order, identified via its second (total) order with permutations, is a Hopf algebra homomorphism, which is isometric and preserves the internal product, onto the Hopf algebra of permutations, previously considered by the two authors. Finally, the scalar product between any special double poset and double posets naturally associated to integer partitions is described by an extension of the Littlewood-Richardson rule.

We strengthen and put in a broader perspective previous results of the first two authors on colliding permutations. The key to the present approach is a new non-asymptotic invariant for graphs.