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Register a new userThe Turan Number of the Triangular Pyramid of $3$-Layers
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The Turan number of a graph $H$, denoted by $text{ex}(n, H)$, is the maximum number of edges in an $n$-vertex graph that does not have $H$ as a subgraph. Let $TP_k$ be the triangular pyramid of $k$-layers. In this paper, we determine that $text{ex}(n,TP_3)= frac{1}{4}n^2+n+o(n)$ and pose a conjecture for $text{ex}(n,TP_4)$.
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The Turan 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 $ex(n,P_3cup P_{2ell+1})$ for $ngeq 2ell+4$. 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 $[n,14,7]=(5n+91+r(r-6))/2$, $n-13equiv r,(text{mod }6)$ and $0leq r< 6$.
Let $F$ be a graph. The planar Turan number of $F$, denoted by $text{ex}_{mathcal{P}}(n,F)$, is the maximum number of edges in an $n$-vertex planar graph containing no copy of $F$ as a subgraph. Let $Theta_k$ denote the family of Theta graphs on $kgeq 4$ vertices, that is, a graph obtained by joining a pair of non-consecutive vertices of a $k$-cycle with an edge. Y. Lan, et.al. determined sharp upper bound for $text{ex}_{mathcal{P}}(n,Theta_4)$ and $text{ex}_{mathcal{P}}(n,Theta_5)$. Moreover, they obtained an upper bound for $text{ex}_{mathcal{P}}(n,Theta_6)$. They proved that, $text{ex}_{mathcal{P}}(n,Theta_6)leq frac{18}{7}n-frac{36}{7}$. In this paper, we improve their result by giving a bound which is sharp. In particular, we prove that $text{ex}_{mathcal{P}}(n,Theta_6)leq frac{18}{7}n-frac{48}{7}$ and demonstrate that there are infinitely many $n$ for which there exists a $Theta_6$-free planar graph $G$ on $n$ vertices, which attains the bound.
Let ${rm ex}_{mathcal{P}}(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex planar graph which does not contain $H$ as a subgraph. When $T=K_2$, ${rm ex}_{mathcal{P}}(n,T,H)$ is the well studied function, the planar Turan number of $H$, denoted by ${rm ex}_{mathcal{P}}(n,H)$. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both ${rm ex}_{mathcal{P}}(n,C_4)$ and ${rm ex}_{mathcal{P}}(n,C_5)$. Later on, Y. Lan, et al. continued this topic and proved that ${rm ex}_{mathcal{P}}(n,C_6)leq frac{18(n-2)}{7}$. In this paper, we give a sharp upper bound ${rm ex}_{mathcal{P}}(n,C_6) leq frac{5}{2}n-7$, for all $ngeq 18$, which improves Lans result. We also pose a conjecture on ${rm ex}_{mathcal{P}}(n,C_k)$, for $kgeq 7$.
Let $F$ be a fixed graph. The rainbow Turan number of $F$ is defined as the maximum number of edges in a graph on $n$ vertices that has a proper edge-coloring with no rainbow copy of $F$ (where a rainbow copy of $F$ means a copy of $F$ all of whose edges have different colours). The systematic study of such problems was initiated by Keevash, Mubayi, Sudakov and Verstraete. In this paper, we show that the rainbow Turan number of a path with $k+1$ edges is less than $left(frac{9k}{7}+2right) n$, improving an earlier estimate of Johnston, Palmer and Sarkar.
A hypergraph is linear if any two of its edges intersect in at most one vertex. The Sail (or $3$-fan) $F^3$ is the $3$-uniform linear hypergraph consisting of $3$ edges $f_1, f_2, f_3$ pairwise intersecting in the same vertex $v$ and an additional edge $g$ intersecting all $f_i$ in a vertex different from $v$. The linear Turan number $ex_{lin}(n, F^3)$ is the maximum number of edges in a $3$-uniform linear hypergraph on $n$ vertices that does not contain a copy of $F^3$. F{u}redi and Gyarfas proved that if $n = 3k$, then $ex_{lin}(n, F^3) = k^2$ and the only extremal hypergraphs in this case are transversal designs. They also showed that if $n = 3k+2$, then $ex_{lin}(n, F^3) = k^2+k$, and the only extremal hypergraphs are truncated designs (which are obtained from a transversal design on $3k+3$ vertices with $3$ groups by removing one vertex and all the hyperedges containing it) along with three other small hypergraphs. However, the case when $n =3k+1$ was left open. In this paper, we solve this remaining case by proving that $ex_{lin}(n, F^3) = k^2+1$ if $n = 3k+1$, answering a question of F{u}redi and Gyarfas. We also characterize all the extremal hypergraphs. The difficulty of this case is due to the fact that these extremal examples are rather non-standard. In particular, they are not derived from transversal designs like in the other cases.
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