Summary: H-free graphs of large minimum degree
We prove the following extension of an old result of AndrŽasfai, Erdos and SŽos. For every fixed
graph H with chromatic number r + 1 3, and for every fixed > 0, there are n0 = n0(H, ) and
= (H) > 0, such that the following holds. Let G be an H-free graph on n > n0 vertices with
minimum degree at least 1 - 1
r-1/3 + n. Then one can delete at most n2-
edges to make G
TurŽan's classical Theorem  determines the maximum number of edges in a Kr+1-free graph on n
vertices. It easily implies that for r 2, if a Kr+1-free graph on n vertices has minimum degree at
least (1- 1
r )n, then it is r-colorable (in fact, it is a complete r-partite graph with equal color classes).
The following stronger result has been proved by AndrŽasfai, Erdos and SŽos .
Theorem 1.1 () If G is a Kr+1-free graph of order n with minimum degree (G) > 1 - 1
then G is r-colorable.