# Dense subgraphs in the H-free process

@article{Warnke2011DenseSI,
title={Dense subgraphs in the H-free process},
author={Lutz Warnke},
journal={Discret. Math.},
year={2011},
volume={311},
pages={2703-2707}
}
• L. Warnke
• Published 1 March 2010
• Mathematics
• Discret. Math.
The Reverse H‐free Process for Strictly 2‐Balanced Graphs
Consider the random graph process that starts from the complete graph on n vertices and gives the exact asymptotics on the number of edges remaining in the graph when the process terminates and investigates some basic properties namely the size of the maximal independent set and the presence of subgraphs.
Dynamic concentration of the triangle-free process
• Mathematics
• 2013
The triangle-free process begins with an empty graph on n vertices and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. We determine the
The Final Size of the C4-Free Process
We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of C4. We show that, with probability tending to
The diamond‐free process
This analysis suggests that the graph produced after i edges are added resembles the uniform random graph, with the additional condition that the edges which do not lie on triangles form a random‐looking subgraph.
When does the K4‐free process stop?
• L. Warnke
• Mathematics
Random Struct. Algorithms
• 2014
An analysis of the K4-free process shows that with high probability G has edges and is ‘nearly regular’, i.e., every vertex has degree .
The Cℓ‐free process
• L. Warnke
• Mathematics
Random Struct. Algorithms
• 2014
A rigorous way to 'transfer' certain decreasing properties from the binomial random graph to the H-free process is established, which confirms a conjecture of Bohman and Keevash and improves on bounds of Osthus and Taraz.
J an 2 01 1 The C l-free process
The Cl-free process starts with the empty graph on n vertices and adds edges chosen uniformly at random, one at a time, subject to the condition that no copy of Cl is created. For every l ≥ 4 we show
The Final Size of the Cℓ-free Process
The differential equation method is used and it is shown that the final graph produced by this random graph process has maximum degree $O( (n \log n)^{1/(\ell-1)})$ and, consequently, size $O (n^{\ell/(\ll-1)}(\log n). The Final Size of the Cℓ-free Process The differential equation method is used and it is shown that the final graph produced by this random graph process has maximum degree$O( (n \log n)^{1/(\ell-1)})$and, consequently, size$O (n^{\ell/(\ll-1)}(\log n).

## References

SHOWING 1-10 OF 21 REFERENCES
The early evolution of the H-free process
• Mathematics
• 2009
The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject
Random maximal H-free graphs
• Mathematics
• 2001
Given a graph H, a random maximal H-free graph is constructed by the following random greedy process. First assign to each edge of the complete graph on n vertices a birthtime which is uniformly
No Dense Subgraphs Appear in the Triangle-free Graph Process
• Mathematics
Electron. J. Comb.
• 2011
It is shown that there exists a constant $c$ such that asymptotically almost surely no copy of any fixed finite triangle-free graph on $k$ vertices with at least $ck$ edges appears in the triangle- free graph process.
The K_4-free process
We consider the K_4-free process. In this process, the edges of the complete n-vertex graph are traversed in a uniformly random order, and each traversed edge is added to an initially empty evolving
The Final Size of the C4-Free Process
We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of C4. We show that, with probability tending to
4-cycles at the Triangle-free Process
On the Size of a Random Maximal Graph
• Mathematics
Random Struct. Algorithms
• 1995
A variety of techniques are used to show that the size of the random maximal bipartite graph is quadratic in n but of order only n3'2 in the triangle—free case and a slight improvement in the lower bound of the Ramsey number r(3, t).
The random planar graph process
• Mathematics
Random Struct. Algorithms
• 2008
The following variant of the classical random graph process introduced by Erdős and Rényi is considered, showing that for all ε > 0, with high probability, θ(n) edges have to be tested before the number of edges in the graph reaches (1 + ε)n.
Triangle‐free subgraphs in the triangle‐free process
Consider the triangle‐free process, which is defined as follows. Start with G(0), an empty graph on n vertices. Given G(i ‐ 1), let G(i) = G(i ‐ 1) ∪{g(i)}, where g(i) is an edge that is chosen