Tur\`an numbers of Multiple Paths and Equibipartite Trees

TL;DR

This paper determines the maximum number of edges in large graphs avoiding multiple disjoint paths and equibipartite trees, advancing Turán-type extremal graph theory results.

Contribution

It explicitly calculates Turán numbers for multiple paths and equibipartite forests, confirming a conjecture for certain cases and extending known bounds.

Findings

Determined ex(n, kP_3) for large n.

Extended results to ex(n, kP_l) for arbitrary l.

Conditional results on equibipartite forests based on Erdős–Sós conjecture.

Abstract

The Tur\'an number of a graph H, ex(n;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 kP_l denote k vertex-disjoint copies of P_l. We determine ex(n, kP_3) for n appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex (n, kP_l) for arbitrary l, and n appropriately large relative to k and l. We provide some background on the famous Erd\H{o}s-S\'os conjecture, and conditional on its truth we determine ex(n;H) when H is an equibipartite forest, for appropriately large n.