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

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

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

**Duration**: This information is not available in the database

**Location**: OAT S15/S16/S17

**Speaker**: Lutz Warnke

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.

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