Summary: H-Factors in Dense Graphs
Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
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.
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 H-factor if it contains n vertex disjoint copies of
H. Thus, for example, a K2-factor is simply a perfect matching, whereas a C4-factor 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