Summary: Refining the Graph Density Condition for the Existence of Almost
Alon and Yuster  have proven that if a fixed graph K on g vertices is (h + 1)-colorable, then any
graph G with n vertices and minimum degree at least h
h+1 n contains at least (1 - )n
g vertex disjoint
copies of K, provided n > N( ). It is shown here that the required minimum degree of G for this
result to follow is closer to h-1
h n, provided K has a proper (h + 1)-coloring in which some of the colors
occur rarely. A conjecture regarding the best possible result of this type is suggested.
For an infinite family of graphs F and a fixed graph K with g vertices, we say that the graphs in F
contain an almost K-factor if for any > 0 there exists an N = N( ) such that if G is a graph from F
with n > N vertices, then G contains at least (1 - )n
g vertex disjoint copies of K.
When K is characterized only by its chromatic number, say h+1, it is sufficient to consider a complete
(h + 1)-partite graph, Ka0,...,ah