UNIVERSAL GRAPHS AND UNIVERSAL PERMUTATIONS

Universal cycles for permutations

A universal cycle for permutations is a word of length n! such that each of the n! possible relative orders of n distinct integers occurs as a cyclic interval of the word. We show how to construct such a universal cycle in which only n + 1 distinct integers are used. This is best possible and proves a conjecture of Chung, Diaconis and Graham.

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FLOWS AND UNIVERSAL COMPACTIFICATIONS

The main purpose of this paper is to establish a relation between universality of certain P-compactifications of a semitopological semigroup and their corresponding enveloping semigroups. In particular, we show that if we take P to be the property that the enveloping semigroup of a compactification of a semitopological semigroup s is left simple, a group, or the trivial singleton semigroup, t...

Sparse universal graphs

For every n, we describe an explicit construction of a graph on n vertices with at most O(n2− ) edges, for = 0.133 . . ., that contains every graph on n vertices with maximum degree 3 as a subgraph. It is easy to see that each such graph must have at least Ω(n 4 3 ) edges. We also show that the minimum number of edges of a graph that contains every graph with n edges as a subgraph is Θ( n 2 log...

K-universal Finite Graphs

This paper investigates the class of k universal nite graphs a local analog of the class of universal graphs which arises naturally in the study of nite variable logics The main results of the paper which are due to Shelah establish that the class of k universal graphs is not de nable by an in nite disjunction of rst order existential sentences with a nite number of variables and that there exi...