## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Monday, September 07, 2009, 12:15 pm

Duration: This information is not available in the database

Location: CAB G51

Speaker: Lutz Warnke

## On the Evolution of K_{\ell+1}-free Graphs

In this talk we discuss K_{\ell+1}-free graphs (i.e. graphs which do not contain K_{\ell+1} subgraphs), where \ell \geq 2 is some fixed integer. Kolaitis, Prömel and Rothschild (1986) proved that a random K_{\ell+1}-free graph on n vertices is almost surely \ell-colorable. Is a similar result true for random K_{\ell+1}-free graphs on n vertices having exactly m=m(n) edges?

We show that the evolution of random K_{\ell+1}-free graphs exhibits two phase transitions with respect to being \ell-colorable as m increases from 0 to n^2: first it is almost surely \ell-colorable, then it is not, and then it is once again. In particular, we prove an explicit threshold function t_\ell=t_\ell(n), such that a random K_{\ell+1}-free graph with m > t_\ell edges is almost surely \ell-colorable whereas for c n \leq m < t_\ell edges it almost surely has chromatic number at least \ell+1 for an appropriate constant c=c(\ell). Our results hold provided the so-called KLR-Conjecture is true for K_{\ell+1}. The latter has been verified for K_3, K_4 and K_5, i.e. for \ell \leq 4.

This extends results of Osthus, Prömel and Taraz (2003) as well as Steger (2005), who proved the described phenomenon for triangle-free graphs, i.e. for the special case \ell=2.

Joint work with Angelika Steger.

Previous talks by year:   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