The Turan number of 2P_7

Abstract

The Turán 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 $m{P}_k$ denote $m$ disjoint copies of $P_k$. Bushaw and Kettle [Turán numbers of multiple paths and equibipartite forests, Combin. Probab. Comput. 20(2011) 837--853] determined the exact value of $ex(n,k{P}_{\ell })$ for large values of $n$. Yuan and Zhang [The Turán number of disjoint copies of paths, Discrete Math. 340(2)(2017) 132--139] completely determined the value of $ex(n,k{P}_3)$ 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 $n\ge 2\ell +4$. Recently, Bielak and Kieliszek [The Turán number of the graph $2P_5$, Discuss. Math. Graph Theory 36(2016) 683--694], Yuan and Zhang [Turán 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 $0\le r<6$.

9 pages