 
Summary: Refining the Graph Density Condition for the Existence of Almost
Kfactors
Noga Alon
Eldar Fischer
Abstract
Alon and Yuster [4] 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 h1
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.
1 Introduction
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 Kfactor 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
