Allen, Peter (2010) Dense H-free graphs are almost (χ(H)−1)-partite. The electronic journal of combinatorics, 27 (R21). ISSN 1077-8926
By using the Szemerédi Regularity Lemma, Alon and Sudakov recently extended the classical Andrásfai-Erdős-Sós theorem to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any (r+1)-partite graph H whose smallest part has t vertices, there exists a constant C such that for any given ε>0 and sufficiently large n the following is true. Whenever G is an n-vertex graph with minimum degree δ(G)≥(1−33r−1+ε)n, either G contains H, or we can delete f(n,H)≤Cn2−1t edges from G to obtain an r-partite graph. Further, we are able to determine the correct order of magnitude of f(n,H) in terms of the Zarankiewicz extremal function.
|Additional Information:||© 2010 The Author|
|Library of Congress subject classification:||Q Science > QA Mathematics|
|Sets:||Departments > Mathematics|
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