Large induced trees in Kr-free graphs

Chromatic number and minimum degree of Kr+1-free graphs

Let (G) be the minimum degree of a graph G: A number of results about trianglefree graphs determine the maximum chromatic number of graphs of order n with (G) n=3: In this paper these results are extended to Kr+1-free graphs of order n with (G) (1 2= (2r 1))n: In particular: (a) there exist Kr+1-free graphs of order n with (G) > (1 2= (2r 1))n o (n) and arbitrary large chromatic number; (b) if ...

Dense graphs with small clique number

We consider the structure of Kr-free graphs with large minimum degree, and show that such graphs with minimum degree δ > (2r − 5)n/(2r − 3) are homomorphic to the join Kr−3 ∨ H where H is a triangle-free graph. In particular this allows us to generalize results from triangle-free graphs and show that Kr-free graphs with such minimum degree have chromatic number at most r+ 1. We also consider th...

Ks-Free Graphs Without Large Kr-Free Subgraphs

Subgraphs in vertex neighborhoods of Kr-free graphs

The neighbourhood of every vertex of a triangle-free graph forms an independent set. In 1981, Janos Pach characterized those triangle-free graphs where every independent set belongs to the neighbourhood of a vertex. I will present alternative characterizations and indicate some applications of this result. There are two natural ways to generalize this problem to Kr-free graphs: Characterize tho...

Strong Turán Stability

We study maximal Kr+1-free graphs G of almost extremal size—typically, e(G) = ex(n,Kr+1) − O(n). We show that any such graph G must have a large amount of ‘symmetry’: in particular, all but very few vertices of G must have twins. (Two vertices u and v are twins if they have the same neighbourhood.) As a corollary, we obtain a new, short proof of a theorem of Simonovits on the structure of extre...