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

Topics from this paper

A Limit Law of Almost l-partite Graphs
  • Vera Koponen
  • Computer Science, Mathematics
  • The Journal of Symbolic Logic
  • 2013
TLDR
It is proved that for every first-order sentence φ, the proportion of graphs in P n (l, d) that satisfy φ converges as n → ∞, and there is a unique partition of {1, …, n} into l parts such that every vertex has at most d neighbours in its own part. Expand
The structure of almost all graphs in a hereditary property
TLDR
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
An asymmetric container lemma and the structure of graphs with no induced $4$-cycle
The method of hypergraph containers, introduced recently by Balogh, Morris, and Samotij, and independently by Saxton and Thomason, has proved to be an extremely useful tool in the study of variousExpand
The number of Ks, t-free graphs
TLDR
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
Almost all triple systems with independent neighborhoods are semi-bipartite
TLDR
This work proves that almost all triple systems with vertex sets [n] and independent neighborhoods are semi-bipartite, and uses the Frankl-Rodl hypergraph regularity lemma, and stability theorems to do so. Expand
The number of $C_{2l}$-free graphs
One of the most basic questions one can ask about a graph $H$ is: how many $H$-free graphs on $n$ vertices are there? For non-bipartite $H$, the answer to this question has been well-understood sinceExpand
The Number of Triple Systems Without Even Cycles
TLDR
For every even integer k ⩾ 4, there exists c > 0 such that the number of triple systems with vertex set [n] containing no Ck is at most $$2^{cn^2}$$2cn2. Expand
The typical structure of sparse $K_{r+1}$-free graphs
Two central topics of study in combinatorics are the so-called evolution of random graphs, introduced by the seminal work of Erd\H{o}s and R\'enyi, and the family of $H$-free graphs, that is, graphsExpand
The number of Km,m-free graphs
TLDR
It is proved that fn(Km,m) ≤ 2O(n2−1/m) for every m, extending the result of Kleitman and Winston and answering a question of Erdős. Expand
Balanced supersaturation for some degenerate hypergraphs
We prove balanced supersaturation theorems for θa,b (the graph consisting of a internally vertex-disjoint paths of length b, each with the same endpoints) and the complete r-partite r-uniformExpand
...
1
2
3
...

References

SHOWING 1-10 OF 35 REFERENCES
On the existence of triangulated spheres in 3-graphs, and related problems
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 ofExpand
Compactness results in extremal graph theory
TLDR
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
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) forExpand
The structure of almost all graphs in a hereditary property
TLDR
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
TLDR
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
TLDR
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
TLDR
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
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 graphExpand
The asymptotic number of graphs not containing a fixed color-critical subgraph
TLDR
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
TLDR
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
...
1
2
3
4
...