On the Turán Number of Forests

Abstract

The Turán number of a graph $H$, $\mathrm{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices which does not have $H$ as a subgraph. We determine the Turán number and find the unique extremal graph for forests consisting of paths when $n$ is sufficiently large. This generalizes a result of Bushaw and Kettle [Combinatorics, Probability and Computing 20:837--853, 2011]. We also determine the Turán number and extremal graphs for forests consisting of stars of arbitrary order.