Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Thursday, December 04, 2014, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Rajko Nenadov

Almost-spanning universality in random graphs

A graph $G$ is said to be $\mathcal{H}(N, \Delta)$-universal if it contains a copy of every graph on $N$ vertices with maximum degree at most $\Delta$. Determining the threshold for the property that a typical graph $G \sim G(n,p)$ is $\mathcal{H}(N, \Delta)$-universal is an intriguing question in the theory of random graphs, with two most common scenarios being $N = n$ (spanning subgraphs) and $N = (1 - \varepsilon)n$ (almost-spanning subgraphs). A result of Alon, Capalbo, Kohayakawa, Rödl, Ruciński and Szemerédi shows that for $N = (1 - \varepsilon)n$ it suffices to take $p \ge (\log n / n)^{1/\Delta}$. This was further improved by Dellamonica, Kohayakawa, Rödl and Rucińcki, who showed that for the same value of $p$ one can take $N$ to be as large as $n$. On the other hand, the only known lower bound on the threshold for these two properties is of order $n^{-2/(\Delta + 1)}$. It is worth noting that even for the simpler property of containing a single (arbitrary) spanning graph $H \in \mathcal{H}(n, \Delta)$, no better bound on $p$ is known (result of Alon and Füredi).

We make a step towards closing this gap. In particular, we improve the result of Alon et al. by showing that, in the case $\Delta \ge 3$, a typical graph $G \sim G(n, p)$ is $\mathcal{H}((1 - \varepsilon)n, \Delta)$-universal for $p \ge n^{-1/(\Delta - 1)} \log^5 n$. This determines, up to the logarithmic factor, the asymptotic value of the threshold in case $\Delta = 3$. Using similar ideas, we also show that there exists a constant $\delta > 0$ such that for any graph $H \in \mathcal{H}(n, \Delta)$ and $p \ge n^{-1/\Delta - \delta}$, a typical graph $G \sim G(n,p)$ contains a copy of $H$. This gives a slight improvement over the result of Alon and Füredi.

Joint work with David Conlon, Asaf Ferber and Nemanja Škorić.

Upcoming talks     |     All previous talks     |     Talks by speaker     |     Upcoming talks in iCal format (beta version!)

Previous talks by year:   2024  2023  2022  2021  2020  2019  2018  2017  2016  2015  2014  2013  2012  2011  2010  2009  2008  2007  2006  2005  2004  2003  2002  2001  2000  1999  1998  1997  1996  

Information for students and suggested topics for student talks

Automatic MiSe System Software Version 1.4803M   |   admin login