Regular pairs behave like complete bipartite graphs from the point of view of bounded degree subgraphs.
In this paper we prove the following conjecture of Alon and Yuster. Let H be a graph with h vertices and chromatic number k. There exist constants c(H) and n0(H) such that if n¿n0(H) and G is a graph with hn vertices and minimum degree at least (1 − 1=k)hn + c(H), then G contains an H-factor. In fact, we show that if H has a k-coloring with color-class… (More)
The purpose of this paper is to describe a sorting network of size 0(n log n) and depth 0(log n). A natural way of sorting is through consecutive halvings: determine the upper and lower halves of the set, proceed similarly within the halves, and so on. Unfortunately, while one can halve a set using only 0(n) comparisons, this cannot be done in less than… (More)
We disprove Heilbronn's conjecture—that N points lying in the unit disc necessarily contain a triangle of area less than c/N 2 .