TY - JOUR

T1 - The fine structure of octahedron-free graphs

AU - Balogh, József

AU - Bollobás, Béla

AU - Simonovits, Miklós

N1 - Funding Information:
E-mail addresses: jobal@math.uiuc.edu (J. Balogh), bb12@cam.ac.uk (B. Bollobás), miki@renyi.hu (M. Simonovits). 1 This material is based upon work supported by NSF CAREER Grants DMS-0745185 and DMS-0600303, UIUC Campus Research Board Grants 06139, 07048 and 08086, and OTKA grants 049398 and K76099. 2 Partially supported by NSF grants DMS-0505550, DMS-0906634, ITR grants CCR-0225610, CNS-0721983 and CCF-0728928 and DARPA grant F33615-01-C-1900, ARO grant W911NF-06-1-0076. 3 Partially supported by OTKA grants T026069, T038210, T069062.

PY - 2011/3

Y1 - 2011/3

N2 - 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 LεL. 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.

AB - 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 LεL. 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.

KW - Extremal graphs

KW - Graph counting

KW - Structure of H-free graphs

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U2 - 10.1016/j.jctb.2010.11.001

DO - 10.1016/j.jctb.2010.11.001

M3 - Article

AN - SCOPUS:78951485003

VL - 101

SP - 67

EP - 84

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 2

ER -