The fine structure of octahedron-free graphs

This paper is one of a series of papers in which, for a family L of graphs, we describe the typical structure of graphs not containing any L ∈ L. In this paper, we prove sharp results about the case L = {O6}, where O6 is the graph with 6 vertices and 12 edges, given by the edges of an octahedron. Among others, we prove the following results. (a) The vertex set of almost every O6-free graph can be partitioned into two classes of almost equal sizes, U1 and U2, where the graph spanned by U1 is a C4-free and that by U2 is P3-free. (b) Similar assertions hold when L is the family of all graphs with 6 vertices and 12 edges. (c) If H is a graph with a color-critical edge and χ(H) = p + 1, then almost every sH-free graph becomes p-chromatic after the deletion of some s−1 vertices, where sH is the graph formed by s vertex disjoint copies of H. These results are natural extensions of theorems of classical extremal graph theory. To show that results like those above do not hold in great generality, we provide examples for which the analogs of our results do not hold.