The Turan number of 2P_7

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 $0\leq r< 6$.