The Turań Number of 2P7

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 Pk denote the path on k vertices and let mPk denote m disjoint copies of Pk. 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, kPℓ) for large values of n. Yuan and Zhang [The Turán number of disjoint copies of paths, Discrete Math. 340 (2017) 132–139] completely determined the value of ex(n, kP3) for all n, and also determined ex(n, Fm), where Fm is the disjoint union of m paths containing at most one odd path. They also determined the exact value of ex(n, P3 ∪ P2ℓ+1) for n ≥ 2ℓ + 4. Recently, Bielak and Kieliszek [The Turán number of the graph 2P5, 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, 2P5). In this paper, we show that ex(n, 2P7) = max{[n, 14, 7], 5n − 14} for all n ≥ 14, where [n, 14, 7] = (5n + 91 + r(r − 6))/2, n − 13 ≡ r (mod 6) and 0 ≤ r < 6.

Keywords

• turán number
• extremal graphs
• 2p7
• 05c35