The fine structure of octahedron-free graphs

  • Authors:

  • József Balogh;Béla Bollobás;Miklós Simonovits

  • Affiliations:

  • Department of Mathematics, University of Illinois, Urbana, IL 61801, USA and Department of Mathematics, U.C. California at San Diego, 9500 Gilmann Drive, La Jolla, USA;Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB, UK and The University of Memphis, Memphis, TN 38152, USA;Rényi Institute, Budapest, Hungary

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2011

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Abstract

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={O"6}, where O"6 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 O"6-free graph can be partitioned into two classes of almost equal sizes, U"1 and U"2, where the graph spanned by U"1 is a C"4-free and that by U"2 is P"3-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 @g(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.