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 h h+1n 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.

[1]  Noga Alon,et al.  AlmostH-factors in dense graphs , 1992, Graphs Comb..

[2]  C. V. Eynden,et al.  A proof of a conjecture of Erdös , 1969 .

[3]  János Komlós,et al.  Proof of a Packing Conjecture of Bollobás , 1995, Combinatorics, Probability and Computing.

[4]  Mohamed H. El-Zahar,et al.  On circuits in graphs , 1984, Discret. Math..

[5]  Noga Alon,et al.  2-factors in Dense Graphs , 1996, Discret. Math..

[6]  M. Simonovits,et al.  Szemeredi''s Regularity Lemma and its applications in graph theory , 1995 .

[7]  Vojtech Rödl,et al.  The Algorithmic Aspects of the Regularity Lemma , 1994, J. Algorithms.

[8]  M. Aigner,et al.  Embedding Arbitrary Graphs of Maximum Degree Two , 1993 .

[9]  Noga Alon,et al.  AlmostH-factors in dense graphs , 1992, Graphs Comb..

[10]  E. Szemerédi.  Regular Partitions of Graphs , 1975 .

[11]  János Komlós,et al.  On the Pósa-Seymour conjecture , 1998, J. Graph Theory.