Related papers: On hypergraphs without loose cycles

We prove an upper bound for the number of edges a C4-free graph on q^2 + q vertices can contain for q even. This upper bound is achieved whenever there is an orthogonal polarity graph of a plane of even order q.

For a 3-uniform hypergraph (3-graph) $F$, let $r(F,n)$ be the smallest $N$ such that any $N$-vertex $F$-free 3-graph has an independent set of size $n$. We construct a $3$-graph $H_2$ with six vertices and five edges such that…

Popielarz, Sahasrabudhe and Snyder in 2018 proved that maximal $K_{r+1}$-free graphs with $(1-\frac{1}{r})\frac{n^2}{2}-o(n^{\frac{r+1}{r}})$ edges contain a complete $r$-partite subgraph on $n-o(n)$ vertices. This was very recently…

A Berge cycle of length $\ell$ in a hypergraph is an alternating sequence of $\ell$ distinct vertices and $\ell$ distinct edges $v_1,e_1,v_2, \ldots, v_\ell, e_{\ell}$ such that $\{v_i, v_{i+1}\} \subseteq e_i$ for all $i$, with indices…

In this paper, we study the problem of determining the maximum number of edges in an $n$-vertex $r$-uniform hypergraph that contains no $(k+1)$-connected subgraph. The graph case is a classical problem initiated by Mader, central to graph…

The suspension of the path $P_4$ consists of a $P_4$ and an additional vertex connected to each of the four vertices, and is denoted by $\hat{P_4}$. The largest number of triangles in a $\hat{P_4}$-free $n$-vertex graph is denoted by…

Given $k\ge3$ and $1\leq \ell< k$, an $(\ell,k)$-cycle is one in which consecutive edges, each of size $k$, overlap in exactly $\ell$ vertices. We study the smallest number of edges in $k$-uniform $n$-vertex hypergraphs which do not contain…

For $k \geq 3$, we prove (i) there is a finite number of $k$-vertex-critical $(P_2+\ell P_1)$-free graphs and (ii) $k$-vertex-critical $(P_3+P_1)$-free graphs have at most $2k-1$ vertices. Together with previous research, these results…

A conjecture of Birmel\'e, Bondy and Reed states that for any integer $\ell\geq 3$, every graph $G$ without two vertex-disjoint cycles of length at least $\ell$ contains a set of at most $\ell$ vertices which meets all cycles of length at…

A well-known result of Ajtai et al. from 1982 states that every $k$-graph $H$ on $n$ vertices, with girth at least five, and average degree $t^{k-1}$ contains an independent set of size $c n (\log t)^{1/(k-1)}/t$ for some $c>0$. In this…

We construct a countable infinite graph G that does not contain cycles of length four having the property that the sequence of graphs $G_n$ induced by the first $n$ vertices has minimum degree $\delta(G_n)> n^{\sqrt{2}-1+o(1)}$.

An embedding of a graph in $3$-space is linkless if for every two disjoint cycles there exists an embedded ball that contains one of the cycles and is disjoint from the other. We prove that every bipartite linklessly embeddable (simple)…

The $(n-\ell)$-deck of an $n$-vertex graph is the multiset of subgraphs obtained from it by deleting $\ell$ vertices. A family of $n$-vertex graphs is $\ell$-recognizable if every graph having the same $(n-\ell)$-deck as a graph in the…

Recently Lin, Wang and Zhou have proved that every $3$-connected nonbipartite graph of minimum degree at least $k$ with $k\ge 6$ and order at least $k+2$ contains $k$ cycles of consecutive lengths. They also conjecture that this result is…

If $G$ is a claw-free hamiltonian graph of order $n$ and maximum degree $\Delta$ with $\Delta\geq 24$, then $G$ has cycles of at least $\min\left\{ n,\left\lceil\frac{3}{2}\Delta\right\rceil\right\}-2$ many different lengths.

We prove for all $k\geq 4$ and $1\leq\ell<k/2$ the sharp minimum $(k-2)$-degree bound for a $k$-uniform hypergraph $\mathcal H$ on $n$ vertices to contain a Hamiltonian $\ell$-cycle if $k-\ell$ divides $n$ and $n$ is sufficiently large.…

The maximum size of an $r$-uniform hypergraph without a Berge cycle of length at least $k$ has been determined for all $k \ge r+3$ by F\"uredi, Kostochka and Luo and for $k<r$ (and $k=r$, asymptotically) by Kostochka and Luo. In this paper,…

A classical result of Bondy and Simonovits in extremal graph theory states that if a graph on $n$ vertices contains no cycle of length $2k$ then it has at most $O(n^{1+1/k})$ edges. However, matching lower bounds are only known for…

In this note we obtain upper bounds on the number of hyperedges in 3-uniform hypergraphs not containing a Berge cycle of given odd length. We improve the bound given by F\"uredi and \"Ozkahya in 2017. The result follows from a more general…

An $r$-uniform hypergraph is called an $r$-graph. A hypergraph is linear if every two edges intersect in at most one vertex. Given a linear $r$-graph $H$ and a positive integer $n$, the linear Tur\'an number $ex_L(n,H)$ is the maximum…