 
Summary: HFactors in Dense Graphs
Noga Alon
and
Raphael Yuster
Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
Abstract
The following asymptotic result is proved. For every > 0, and for every positive integer h,
there exists an n0 = n0( , h) such that for every graph H with h vertices and for every n > n0,
any graph G with hn vertices and with minimum degree d ((H)1
(H) + )hn contains n vertex
disjoint copies of H. This result is asymptotically tight and its proof supplies a polynomial time
algorithm for the corresponding algorithmic problem.
1 Introduction
All graphs considered here are finite, undirected and simple. If H is a graph on h vertices and G
is a graph on hn vertices, we say that G has an Hfactor if it contains n vertex disjoint copies of
H. Thus, for example, a K2factor is simply a perfect matching, whereas a C4factor is a spanning
subgraph of G every connected component of which is a cycle of length 4.
Let H be a graph on h vertices and let G be a graph on hn vertices. There are several known
