Turán numbers of Multiple Paths and Equibipartite Trees
Neal Bushaw, University of Memphis
Special time/location:Thursday, October 13, 2011, 2:30-3:30pm, Oldfather 205
Abstract:
The Turán number of a graph \(H\) is the maximum number of edges in any graph on \(n\) vertices which does not contain \(H\) as a subgraph. Let \(P_l\) denote a path on \(l\) vertices, and \(k\cdot P_l\) denote \(k\) vertex disjoint copies of \(P_l\). We first determine \(ex(n,k\cdot~P_3)\), answering in the positive a conjecture of Gorgol. Further, we determine \(ex\left(n,k\cdot P_l\right)\) for arbitrary \(l\), and \(n\) appropriately large relative to \(k\) and \(l\). We provide a some background on the famous Erdős-Sós conjecture, and conditional on its truth we obtain tight bounds on \(ex(n,H)\) when \(H\) is a forest consisting of equibipartite trees, for appropriately large \(n\). This is joint work with Nathan Kettle at the University of Cambridge.