On the Turan number of forests

TL;DR

This paper determines the maximum number of edges in large graphs that avoid certain forest subgraphs, specifically paths and stars, and characterizes the extremal graphs for these cases.

Contribution

It generalizes previous results by explicitly finding Turan numbers and extremal graphs for forests of paths and stars, extending known bounds.

Findings

Exact Turan numbers for forests of paths and stars

Unique extremal graphs identified for large n

Generalization of previous results by Bushaw and Kettle

Abstract

The Turan number of a graph H, 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 Turan 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 Turan number and extremal graphs for forests consisting of stars of arbitrary order.