# The fine structure of octahedron-free graphs

@article{Balogh2011TheFS, title={The fine structure of octahedron-free graphs}, author={J{\'o}zsef Balogh and B{\'e}la Bollob{\'a}s and Mikl{\'o}s Simonovits}, journal={J. Comb. Theory, Ser. B}, year={2011}, volume={101}, pages={67-84} }

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 [email protected]?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… Expand

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#### References

SHOWING 1-10 OF 36 REFERENCES

On the existence of triangulated spheres in 3-graphs, and related problems

- Mathematics
- 1973

1. An r-graph, H (') consists of a set T7(H ( ') ) o I' vertices, and a class E(H( ' ) of r-subsets of V(H( ' ) ) (i .e . unordered r-tuples of vertices) . We shall use various letters (in place of… Expand

Compactness results in extremal graph theory

- Mathematics, Computer Science
- Comb.
- 1982

The main purpose of this paper is to prove some compactness results for the case when L consists of cycles, and one of the main tools will be finding lower bounds on the number of pathsPk+1 in a graph ofn vertices andE edges. Expand

The number of graphs without forbidden subgraphs

- Mathematics, Computer Science
- J. Comb. Theory, Ser. B
- 2004

Given a family L of graphs, set p = p(L) = minL ⊂ L χ(L) - 1 and, for n ≥ 1, denote by P(n, L) the set of graphs with vertex set [n] containing no member of L as a subgraph, and write ex(n, L) for… Expand

The structure of almost all graphs in a hereditary property

- Computer Science, Mathematics
- J. Comb. Theory, Ser. B
- 2011

The structure of almost every graph in a hereditary property of graphs, P, is described, and essentially optimal bounds on the speed of P are derived, improving the Alekseev-Bollobas-Thomason Theorem and also generalising results of Balogh, Bollobas and Simonovits. Expand

ON A PROBLEM OF GRAPH THEORY

- Computer Science
- 1966

A long-standing problem about the maximal number of edges of a graph not containing a cycle of length 4 is solved and some unsolved problems are mentioned. Expand

The typical structure of graphs without given excluded subgraphs

- Computer Science
- Random Struct. Algorithms
- 2009

The typical structure of L-free graphs is described, improving the earlier results on the ErdősFrankl-Rödl theorem by proving the earlier conjecture that, for p = p(L) = minL∈L χ (L)− 1, the structure of almost all L- free graphs is very similar to that of a random subgraph of the Turán graph Tn. Expand

The number of Ks, t-free graphs

- Computer Science, Mathematics
- J. Lond. Math. Soc.
- 2011

It is shown that for all s and t satisfying 2 ≤ s ≤ t, fn(Ks,t) = 2 O(n), which is asymptotically sharp for those values of s andT for which the order of magnitude of the Turan number ex(n,Ks-t) is known. Expand

Extremal graph problems with symmetrical extremal graphs. Additional chromatic conditions

- Computer Science, Mathematics
- Discret. Math.
- 1974

The main result of this paper is that for a special, but rather wide class of ''sample graphs'', the extremal graphs, i.e. the graphs of n vertices without subgraphs isomorphic to the sample graph… Expand

The asymptotic number of graphs not containing a fixed color-critical subgraph

- Computer Science, Mathematics
- Comb.
- 1992

The class of those graphsH which have the property that almost all graphs inForb(H) are ℓ-colorable are characterized, and it is shown that this class corresponds exactly to the class of graphs whose extremal graph is the Turán-graphTn(ℓ). Expand

Excluding induced subgraphs: Critical graphs

- Mathematics, Computer Science
- Random Struct. Algorithms
- 2011

The concept of H-free to induced subgraph containment is strengthened, proving that if H has coloring number k + 1 then almost all H- free graphs can be covered by k graphs that are cliques or independent sets if and only if H is in some well-defined sense critical. Expand