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The theta graph $Theta_{ell,t}$ consists of two vertices joined by $t$ vertex-disjoint paths of length $ell$ each. For fixed odd $ell$ and large $t$, we show that the largest graph not containing $Theta_{ell,t}$ has at most $c_{ell} t^{1-1/ell}n^{1+1/ell}$ edges and that this is tight apart from the value of $c_{ell}$.
Let $mathrm{rex}(n, F)$ denote the maximum number of edges in an $n$-vertex graph that is regular and does not contain $F$ as a subgraph. We give lower bounds on $mathrm{rex}(n, F)$, that are best possible up to a constant factor, when $F$ is one of
For given graphs $G$ and $F$, the Turan number $ex(G,F)$ is defined to be the maximum number of edges in an $F$-free subgraph of $G$. Foucaud, Krivelevich and Perarnau and later independently Briggs and Cox introduced a dual version of this problem w
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 Turan number $text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{text{multi}})$ $bi
For a graph $H$ consisting of finitely many internally disjoint paths connecting two vertices, with possibly distinct lengths, we estimate the corresponding extremal number $text{ex}(n,H)$. When the lengths of all paths have the same parity, $text{ex
Let the bipartite Turan number $ex(m,n,H)$ of a graph $H$ be the maximum number of edges in an $H$-free bipartite graph with two parts of sizes $m$ and $n$, respectively. In this paper, we prove that $ex(m,n,C_{2t})=(t-1)n+m-t+1$ for any positive int