Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

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

**Duration**: 30 minutes

**Location**: CAB G51

**Speaker**: Rajko Nenadov

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: 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