Refining the Graph Density Condition for the Existence of Almost \(K\)-factors

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 \(\frac{h}{h+1}n\) contains at least \((1-\epsilon)\frac{n}{g})\) vertex disjoint copies of \(K\), provided \(n>N(\epsilon)\). It is shown here that the required minimum degree of \(G\) for this result to follow is closer to \(\frac{h-1}{h }n\), 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.