arXiv subjects

Lutz Warnke

Publications and source records attributed to Lutz Warnke.

39 records · Page 3Linked to original sources

arXiv2012

When does the K_4-free process stop?

The K_4-free process starts with the empty graph on n vertices and at each step adds a new edge chosen uniformly at random from all remaining edges that do not complete a copy of K_4. Let G be the random maximal K_4-free graph obtained at the end of the process. We show that for some positive constant C, with high probability as $n \to \infty$, the maximum degree in G is at most $C n^{3/5}\sqrt[5]{\log n}$. This resolves a conjecture of Bohman and Keevash for the K_4-free process and improves on previous bounds obtained by Bollobás and Riordan and by Osthus and Taraz. Combined with results of Bohman and Keevash this shows that with high probability G has $Θ(n^{8/5}\sqrt[5]{\log n})$ edges and is `nearly regular', i.e., every vertex has degree $Θ(n^{3/5}\sqrt[5]{\log n})$. This answers a question of Erdős, Suen and Winkler for the K_4-free process. We furthermore deduce an additional structural property: we show that whp the independence number of G is at least $Ω(n^{2/5}(\log n)^{4/5}/\log \log n)$, which matches an upper bound obtained by Bohman up to a factor of $Θ(\log \log n)$. Our analysis of the K_4-free process also yields a new result in Ramsey theory: for a special case of a well-studied function introduced by Erdős and Rogers we slightly improve the best known upper bound.

math.CO↗

arXiv2012

The C_\ell-free process

The C_\ell-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 C_\ell is created. For every $\ell \geq 4$ we show that, with high probability as $n \to \infty$, the maximum degree is $O((n \log n)^{1/(\ell-1)})$, which confirms a conjecture of Bohman and Keevash and improves on bounds of Osthus and Taraz. Combined with previous results this implies that the C_\ell-free process typically terminates with $Θ(n^{\ell/(\ell-1)}(\log n)^{1/(\ell-1)})$ edges, which answers a question of Erdős, Suen and Winkler. This is the first result that determines the final number of edges of the more general H-free process for a non-trivial \emph{class} of graphs H. We also verify a conjecture of Osthus and Taraz concerning the average degree, and obtain a new lower bound on the independence number. Our proof combines the differential equation method with a tool that might be of independent interest: we establish a rigorous way to `transfer' certain decreasing properties from the binomial random graph to the H-free process.

math.CO↗

arXiv2011

Dense subgraphs in the H-free process

The H-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 H is created, where H is some fixed graph. When H is strictly 2-balanced, we show that for some c,d>0, with high probability as $n \to \infty$, the final graph of the H-free process contains no subgraphs F on $v_F \leq n^{d}$ vertices with maximum density $\max_{J \subseteq F}\{e_J/v_J\} \geq c$. This extends and generalizes results of Gerke and Makai for the C_3-free process.

math.CO↗