Extremal graphs for the odd prism

The Tur\'an number $\mathrm{ex}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex graph which does not contain $H$ as a subgraph. The Tur\'{a}n number of regular polyhedrons was widely studied in a series of works due to Simonovits. In this paper, we shall present the exact Tur\'{a}n number of the prism $C_{2k+1}^{\square}$, which is defined as the Cartesian product of an odd cycle $C_{2k+1}$ and an edge $K_2$. Applying a deep theorem of Simonovits and a stability result of Yuan [European J. Combin. 104 (2022)], we shall determine the exact value of $\mathrm{ex}(n,C_{2k+1}^{\square})$ for every $k\ge 1$ and sufficiently large $n$, and we also characterize the extremal graphs. Moreover, in the case of $k=1$, motivated by a recent result of Xiao, Katona, Xiao and Zamora [Discrete Appl. Math. 307 (2022)], we will determine the exact value of $\mathrm{ex}(n,C_{3}^{\square} )$ for every $n$ instead of for sufficiently large $n$.Comment: 24 page

The Turán Number of the Graph 2P5

We give the Turán number ex (n, 2P5) for all positive integers n, improving one of the results of Bushaw and Kettle [Turán numbers of multiple paths and equibipartite forests, Combininatorics, Probability and Computing, 20 (2011) 837-853]. In particular we prove that ex (n, 2P5) = 3n−5 for n ≥ 18

The Turán Number of the Graph 2P₅

We give the Turán number ex (n, 2P5) for all positive integers n, improving one of the results of Bushaw and Kettle [Turán numbers of multiple paths and equibipartite forests, Combininatorics, Probability and Computing, 20 (2011) 837-853]. In particular we prove that ex (n, 2P5) = 3n−5 for n ≥ 18

The Turań Number of 2P7

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