An Algorithmic Embedding of Graphs via Perfect Matchings
Recently Komlós, Sárközy, and Szemerédi proved a striking result called the blow-up lemma that, loosely speaking, enables one to embed any bounded degree graph H as a spanning subgraph of an e-regular graph G. The first proof given by Komlós, Sárközy, and Szemerédi was based on a probabilistic argument . Subsequently, they derandomized their approach to provide an algorithmic embedding in . In this paper we give a different proof of the algorithmic version of the blow-up lemma. Our approach is based on a derandomization of a probabilistic proof of the blow-up lemma given in . The derandomization utilizes the Erdös-Selfridge method of conditional probabilities and the technique of pessimistic estimators.
KeywordsBipartite Graph Perfect Match Regular Graph Algorithmic Version Regularity Lemma
Unable to display preview. Download preview PDF.
- 2.N. Alon, V. Rödl, and A. Ruciński, “Perfect matchings in ε-regular graphs”, The Electronic J. of Combin., vol. 5(1) (1998), # R13.Google Scholar
- 5.A. Hajnal and E. Szemerédi, “Proof of a conjecture of Erdös”, Combinatorial Theory and its Applications, vol. II (P. Erdós, A. Rényi, and V.T. Sós eds), Colloq. Math. Soc. J. Bolyai 4, North Holland, Amsterdam, 1970, pp. 601–623.Google Scholar
- 7.S. Janson, T. łuczak, and A. Ruciński, Topics in Random Graphs, Wiley, New York, 1999.Google Scholar
- 10.J. Komlós, G. N. Sárközy and E. Szemerédi “On the Pósa-Seymour conjecture J. Graph Theory”, Journal of Graph Theory, to appearGoogle Scholar
- 11.J. Komlós, G. N. Sárközy and E. Szemerédi “Proof of the Alon-Yuster conjecture” submittedGoogle Scholar
- 13.V. Rödl and A. Ruciński, “Perfect matchings in ε-regular graphs and the blow-up lemma”, submitted.Google Scholar
- 15.E. Szemerédi “Partitions of graphs” Problems Combin. et Theorie des graphes, Edition du C.N.R.S. vol. 260 (1978), pp. 399–402.Google Scholar