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

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**Date and Time**: Thursday, June 14, 2018, 12:15 pm

**Duration**: 30 minutes

**Location**: CAB G51

**Speaker**: Rajko Nenadov

The Hajnal-Szemerédi theorem states that every graph G with n vertices and minimum degree at least (r-1)n/r contains a K_r-factor, that is a collection of disjoint copies of the complete graph on r vertices which together cover all the vertices of G, assuming r divides n. The bound on the minimum degree is easily seen to be optimal: take a complete r-partite graph in which one vertex class is of size n/r -1, one n/r + 1 and all other of size n/r. Such a graph has minimum degree (r-1)n/r - 1 and clearly does not contain K_r-factor. One particular feature of this example is that it contains a very large independent set. What happens if we exclude graphs with this property? Does a weaker bound on the minimum degree imply a K_r-factor if we additionally assume that G has sublinear independence number? In this talk I will give an overview of known results, present some new ones and discuss further research directions. Based on a joint work with Yanitsa Pehova (University of Warwick).

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