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In 1952, Dirac proved that every $2$-connected $n$-vertex graph with the minimum degree $k+1$ contains a cycle of length at least $\min\{n, 2(k+1)\}$. Here we obtain a stability version of this result by characterizing those graphs with minimum degree $k$ and circumference at most $2k+1$. We present applications of the above-stated result by obtaining generalized Tur\'an numbers. In particular, for all $\ell \geq 5$ we determine how many copies of a five-cycle as well as four-cycle are necessary to guarantee that the graph has circumference larger than $\ell$. In addition, we give a new proof of Luo's Theorem for cliques using our stability result.

In this work, we give the sharp upper bound for the number of cliques in graphs with bounded odd circumferences. This generalized Tur\'an-type result is an extension of the celebrated Erd\H{o}s and Gallai theorem and a strengthening of Luo's recent result. The same bound for graphs with bounded even circumferences is a trivial application of the theorem of Li and Ning.

Given a graph $F$, the planar Tur\'an number of $F$, denoted $\text{ex}_{\mathcal{P}}(n, F)$, is the maximum number of edges in an $n$-vertex $F$-free planar graph. Such an extremal graph problem was initiated by Dowden while determining sharp upper bound for $\text{ex}_{\mathcal{P}}(n,C_4)$ and $\text{ex}_{\mathcal{P}}(n,C_5)$, where $C_4$ and $C_5$ are cycles of length four and five respectively. In this paper we determined an upper bound for $\text{ex}_{\mathcal{P}}(n,S_{2,2})$, $\text{ex}_{\mathcal{P}}(n,S_{2,3})$, $\text{ex}_{\mathcal{P}}(n,S_{2,4})$, $\text{ex}_{\mathcal{P}}(n,S_{2,5})$, $\text{ex}_{\mathcal{P}}(n,S_{3,3})$ and $\text{ex}_{\mathcal{P}}(n,S_{3,4})$, where $S_{m,n}$ is a double star with $m$ and $n$ leafs. Moreover, the bounds for $\text{ex}_{\mathcal{P}}(n,S_{2,2})$ and $\text{ex}_{\mathcal{P}}(n,S_{2,3})$ are sharp.

Let $H$ be a graph. The generalized outerplanar Tur\'an number of $H$, denoted by $f_{\mathcal{OP}}(n,H)$, is the maximum number of copies of $H$ in an $n$-vertex outerplanar graph. Let $P_k$ be the path on $k$ vertices. In this paper we give an exact value of $f_{\mathcal{OP}}(n,P_4)$ and a best asymptotic value of $f_{\mathcal{OP}}(n,P_5)$. Moreover, we characterize all outerplanar graphs containing $f_{\mathcal{OP}}(n,P_4)$ copies of $P_4$., Comment: 23 pages, 12 figures

The Tur\'an 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 $\text{ex}(n,TP_4)$.

One of the first problems in extremal graph theory was the maximum number of edges that a triangle-free graph can have. Mantel solved this, which built the foundations of what we know as extremal graph theory. The natural progression was to ask the maximum number of edges in a $k$-book free graph. A $k$-book, denoted by $B_k$, is $k$ triangles sharing a common edge. In the early 2000s, Gy\H{o}ri \cite{gyori1} solved the hypergraph analog of the maximum number of hyperedges in a triangle-free hypergraph. In a hypergraph, $k$-book denotes $k$ Berge triangles sharing a common edge. Let $\text{ex}_3(n,\mathcal{F})$ denote the maximum number of hyperedges in a Berge-$\mathcal{F}$-free hypergraph on $n$ vertices. In this paper, we prove $\text{ex}_3(n,B_k)=\frac{n^2}{8}(1+o(1))$.

The Tur��n 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 $\text{ex}(n,TP_4)$.

Given a graph $F$, the planar Tur��n number of $F$, denoted $\text{ex}_{\mathcal{P}}(n, F)$, is the maximum number of edges in an $n$-vertex $F$-free planar graph. Such an extremal graph problem was initiated by Dowden while determining sharp upper bound for $\text{ex}_{\mathcal{P}}(n,C_4)$ and $\text{ex}_{\mathcal{P}}(n,C_5)$, where $C_4$ and $C_5$ are cycles of length four and five respectively. In this paper we determined an upper bound for $\text{ex}_{\mathcal{P}}(n,S_{2,2})$, $\text{ex}_{\mathcal{P}}(n,S_{2,3})$, $\text{ex}_{\mathcal{P}}(n,S_{2,4})$, $\text{ex}_{\mathcal{P}}(n,S_{2,5})$, $\text{ex}_{\mathcal{P}}(n,S_{3,3})$ and $\text{ex}_{\mathcal{P}}(n,S_{3,4})$, where $S_{m,n}$ is a double star with $m$ and $n$ leafs. Moreover, the bounds for $\text{ex}_{\mathcal{P}}(n,S_{2,2})$ and $\text{ex}_{\mathcal{P}}(n,S_{2,3})$ are sharp.

We call a $4$-cycle in $K_{n_{1}, n_{2}, n_{3}}$ multipartite, denoted by $C_{4}^{\text{multi}}$, if it contains at least one vertex in each part of $K_{n_{1}, n_{2}, n_{3}}$. The Tur\'an number $\text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})$ $\bigg($ respectively, $\text{ex}(K_{n_{1},n_{2},n_{3}},\{C_{3}, C_{4}^{\text{multi}}\})$ $\bigg)$ is the maximum number of edges in a graph $G\subseteq K_{n_{1},n_{2},n_{3}}$ such that $G$ contains no $C_{4}^{\text{multi}}$ $\bigg($ respectively, $G$ contains neither $C_{3}$ nor $C_{4}^{\text{multi}}$ $\bigg)$. We call a $C^{multi}_4$ rainbow if all four edges of it have different colors. The ant-Ramsey number $\text{ar}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})$ is the maximum number of colors in an edge-colored of $K_{n_{1},n_{2},n_{3}}$ with no rainbow $C_{4}^{\text{multi}}$. In this paper, we determine that $\text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})=n_{1}n_{2}+2n_{3}$ and $\text{ar}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})=\text{ex}(K_{n_{1},n_{2},n_{3}}, \{C_{3}, C_{4}^{\text{multi}}\})+1=n_{1}n_{2}+n_{3}+1,$ where $n_{1}\ge n_{2}\ge n_{3}\ge 1.$

The Tur\'an number of a graph $H$, $\text{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices which does not have $H$ as a subgraph. A wheel $W_n$ is an $n$-vertex graph formed by connecting a single vertex to all vertices of a cycle $C_{n-1}$. Let $mW_{2k+1}$ denote the $m$ vertex-disjoint copies of $W_{2k+1}$. For sufficiently large $n$, we determine the Tur\'an number and all extremal graphs for $mW_{2k+1}$. We also provide the Tur\'an number and all extremal graphs for $W^{h}:=\bigcup\limits^m_{i=1}W_{k_i}$ when $n$ is sufficiently large, where the number of even wheels is $h$ and $h>0$.

Let $F$ be a graph. The planar Tur\'an 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 $k\geq 4$ 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., Comment: 23 pages, 19 figures

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 Tur\'an 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 $n\geq 18$, which improves Lan's result. We also pose a conjecture on ${\rm ex}_{\mathcal{P}}(n,C_k)$, for $k\geq 7$., Comment: 27 pages, 17 figures

Let $f(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. The order of magnitude of $f(n,P_k)$, where $P_k$ is a path on $k$ vertices, is $n^{{\lfloor{\frac{k-1}{2}}\rfloor}+1}$. In this paper we determine the asymptotic value of $f(n,P_5)$ and give conjectures for longer paths., 9 pages, 2 figures

The Tur��n number of a graph $H$, $\text{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices which does not have $H$ as a subgraph. A wheel $W_n$ is an $n$-vertex graph formed by connecting a single vertex to all vertices of a cycle $C_{n-1}$. Let $mW_{2k+1}$ denote the $m$ vertex-disjoint copies of $W_{2k+1}$. For sufficiently large $n$, we determine the Tur��n number and all extremal graphs for $mW_{2k+1}$. We also provide the Tur��n number and all extremal graphs for $W^{h}:=\bigcup\limits^m_{i=1}W_{k_i}$ when $n$ is sufficiently large, where the number of even wheels is $h$ and $h>0$.

The Wiener index of a graph $G$, denoted $W(G)$, is the sum of the distances between all pairs of vertices in $G$. ��. Czabarka, et al. conjectured that for an $n$-vertex, $n\geq 4$, simple quadrangulation graph $G$, \begin{equation*}W(G)\leq \begin{cases} \frac{1}{12}n^3+\frac{7}{6}n-2, &\text{ $n\equiv 0~(mod \ 2)$,}\\ \frac{1}{12}n^3+\frac{11}{12}n-1, &\text{ $n\equiv 1~(mod \ 2)$}. \end{cases} \end{equation*} In this paper, we confirm this conjecture.

We call a $4$-cycle in $K_{n_{1}, n_{2}, n_{3}}$ multipartite, denoted by $C_{4}^{\text{multi}}$, if it contains at least one vertex in each part of $K_{n_{1}, n_{2}, n_{3}}$. The Tur��n number $\text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})$ $\bigg($ respectively, $\text{ex}(K_{n_{1},n_{2},n_{3}},\{C_{3}, C_{4}^{\text{multi}}\})$ $\bigg)$ is the maximum number of edges in a graph $G\subseteq K_{n_{1},n_{2},n_{3}}$ such that $G$ contains no $C_{4}^{\text{multi}}$ $\bigg($ respectively, $G$ contains neither $C_{3}$ nor $C_{4}^{\text{multi}}$ $\bigg)$. We call a $C^{multi}_4$ rainbow if all four edges of it have different colors. The ant-Ramsey number $\text{ar}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})$ is the maximum number of colors in an edge-colored of $K_{n_{1},n_{2},n_{3}}$ with no rainbow $C_{4}^{\text{multi}}$. In this paper, we determine that $\text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})=n_{1}n_{2}+2n_{3}$ and $\text{ar}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})=\text{ex}(K_{n_{1},n_{2},n_{3}}, \{C_{3}, C_{4}^{\text{multi}}\})+1=n_{1}n_{2}+n_{3}+1,$ where $n_{1}\ge n_{2}\ge n_{3}\ge 1.$

Let $F$ be a graph. The planar Tur��n 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 $��_k$ denote the family of Theta graphs on $k\geq 4$ 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 $\text{ex}_{\mathcal{P}}(n,��_4)$ and $\text{ex}_{\mathcal{P}}(n,��_5)$. Moreover, they obtained an upper bound for $\text{ex}_{\mathcal{P}}(n,��_6)$. They proved that, $\text{ex}_{\mathcal{P}}(n,��_6)\leq \frac{18}{7}n-\frac{36}{7}$. In this paper, we improve their result by giving a bound which is sharp. In particular, we prove that $\text{ex}_{\mathcal{P}}(n,��_6)\leq \frac{18}{7}n-\frac{48}{7}$ and demonstrate that there are infinitely many $n$ for which there exists a $��_6$-free planar graph $G$ on $n$ vertices, which attains the bound., 23 pages, 19 figures

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 Tur��n 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 $n\geq 18$, which improves Lan's result. We also pose a conjecture on ${\rm ex}_{\mathcal{P}}(n,C_k)$, for $k\geq 7$., 27 pages, 17 figures

The Tur\'an number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. Let P_k be the path with k vertices, the square P^2_k of P_k is obtained by joining the pairs of vertices with distance one or two in P_k. The powerful theorem of Erd\H{o}s, Stone and Simonovits determines the asymptotic behavior of ex(n,P^2_k). In the present paper, we determine the exact value of ex(n,P^2_5) and ex(n,P^2_6) and pose a conjecture for the exact value of ex(n,P^2_k).

The Tur��n number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. Let P_k be the path with k vertices, the square P^2_k of P_k is obtained by joining the pairs of vertices with distance one or two in P_k. The powerful theorem of Erd��s, Stone and Simonovits determines the asymptotic behavior of ex(n,P^2_k). In the present paper, we determine the exact value of ex(n,P^2_5) and ex(n,P^2_6) and pose a conjecture for the exact value of ex(n,P^2_k).

We are given a set $A$ of buyers, a set $B$ of houses, and for each buyer a preference list, i.e., an ordering of the houses. A house allocation is an injective mapping $\tau$ from $A$ to $B$, and $\tau$ is strictly better than another house allocation $\tau'\neq \tau$ if for every buyer $i$, $\tau'(i)$ does not come before $\tau(i)$ in the preference list of $i$. A house allocation is Pareto optimal if there is no strictly better house allocation. Let $s(\tau)$ be the image of $\tau$ (i.e., the set of houses sold in the house allocation $\tau$). We are interested in the largest possible cardinality $f(m)$ of the family of sets $s(\tau)$ for Pareto optimal mappings $\tau$ taken over all sets of preference lists of $m$ buyers. We improve the earlier upper bound on $f(m)$ given by Asinowski, Keszegh and Miltzow by making a connection between this problem and some problems in extremal set theory.