The Turan number of 2P_7

TL;DR

This paper determines the maximum number of edges in large graphs that do not contain two disjoint paths of length 7, providing an exact formula for the Turán number of 2P_7.

Contribution

It establishes the exact Turán number for 2P_7 for all sufficiently large n, extending previous results on disjoint paths.

Findings

Exact formula for ex(n,2P_7) for all n ≥ 14

Introduces a new extremal graph construction for 2P_7

Complements prior work on disjoint paths and Turán numbers

Abstract

The Tur\'an number of a graph $H$, denoted by $ex(n,H)$, is the maximum number of edges in any graph on $n$ vertices which does not contain $H$ as a subgraph. Let $P_{k}$ denote the path on $k$ vertices and let $mP_{k}$ denote $m$ disjoint copies of $P_{k}$. Bushaw and Kettle [Tur\'{a}n numbers of multiple paths and equibipartite forests, Combin. Probab. Comput. 20(2011) 837--853] determined the exact value of $ex(n,kP_\ell)$ for large values of $n$. Yuan and Zhang [The Tur\'{a}n number of disjoint copies of paths, Discrete Math. 340(2)(2017) 132--139] completely determined the value of $ex(n,kP_3)$ for all $n$, and also determined $ex(n,F_m)$, where $F_m$ is the disjoint union of $m$ paths containing at most one odd path. They also determined the exact value of $ex(n,P_3\cup P_{2\ell+1})$ for $n\geq 2\ell+4$. Recently, Bielak and Kieliszek [The Tur\'{a}n number of the graph $2P_5$, Discuss. Math. Graph Theory 36(2016) 683--694], Yuan and Zhang [Tur\'{a}n numbers for disjoint paths, arXiv: 1611.00981v1] independently determined the exact value of $ex(n,2P_5)$. In this paper, we show that $ex(n,2P_{7})=\max\{[n,14,7],5n-14\}$ for all $n \ge 14$, where $[n,14,7]=(5n+91+r(r-6))/2$, $n-13\equiv r\,(\text{mod }6)$ and $0\leq r< 6$.