## 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: Thursday, December 07, 2006, 12:15 pm

Duration: This information is not available in the database

Location: CAB G51

Speaker: Rico Zenklusen

## A bound on the collection of edges in MST's of fixed-size subgraphs

Let G=(V,E,w) be a complete graph with n vertices and distinct positive edge weights w. We fix a constant k \in {1,2,...,n-1}. For any subset of vertices X of V with cardinality k-1 we denote by MST(G\X) the set of edges contained in the minimum spanning tree in the subgraph G[V\X]. Goemans and Vondrak observed that many edges in G have the property that they are not contained in any minimum spanning tree of the form MST(G\X) where X is as described above. More precisely let M_k be the set of edges contained in any minimum spanning tree of the form MST(G\X), i.e. M_k=\cup_{X\subseteq V, |X|=k-1} MST(G\X).
Goemans and Vondrak made the following conjecture: |M_k| <= n*k-0.5*(k+1)*k.
A proof of this conjecture will be presented in this talk.

Joint work with Gregory B. Sorkin and Angelika Steger.

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