The Turan Number of Disjoint Copies of Paths

TL;DR

This paper determines the maximum number of edges in large graphs that do not contain multiple disjoint paths of length three, extending previous results and characterizing all extremal graphs for this problem.

Contribution

It explicitly calculates the Turán number for disjoint copies of P3 and characterizes all extremal graphs, advancing the understanding of forbidden subgraph configurations.

Findings

Calculated ex(n, k·P3) for all n and k.

Characterized all extremal graphs avoiding k disjoint P3.

Extended previous results and confirmed a conjecture.

Abstract

The Tur\'{a}n number of a graph $H$, $ex(n,H)$, is the maximum number of edges in a simple graph of order $n$ which does not contain $H$ as a subgraph. Let $k\cdot P_3$ denote $k$ disjoint copies of a path on $3$ vertices. In this paper, we determine the value $ex(n, k\cdot P_3)$ and characterize all extremal graphs. This extends a result of Bushaw and Kettle [N. Bushaw and N. Kettle, Tur\'{a}n Numbers of multiple and equibipartite forests, Combin. Probab. Comput., 20(2011) 837-853.], which solved the conjecture proposed by Gorgol in [I. Gorgol. Tur\'{a}n numbers for disjoint copies of graphs. {\it Graphs Combin.}, 27 (2011) 661-667.].