Related papers: Spanning tight components in 4-uniform hypergraphs

Any $n$-vertex $3$-graph with minimum codegree at least $\lfloor n/3\rfloor$ must have a spanning tight component, but immediately below this threshold it is possible for no tight component to span more than $\lceil 2n/3\rceil$ vertices.…

We study minimum vertex-degree conditions in 3-uniform hypergraphs for (tight) spanning components and (combinatorial) surfaces. Our main results show that a 3-uniform hypergraph $G$ on $n$ vertices contains a spanning component if…

We prove that for fixed $k$, every $k$-uniform hypergraph on $n$ vertices and of minimum codegree at least $n/2+o(n)$ contains every spanning tight $k$-tree of bounded vertex degree as a sub\-graph. This generalises a well-known result of…

In 1995, Koml\'os, S\'ark\"ozy and Szemer\'edi showed that every large $n$-vertex graph with minimum degree at least $(1/2 + \gamma)n$ contains all spanning trees of bounded degree. We consider a generalization of this result to loose…

A classical result of Koml\'os, S\'ark\"ozy and Szemer\'edi shows that every large $n$-vertex graph with minimum degree at least $(1/2+\gamma)n$ contains all spanning trees of bounded degree. We generalised this result to loose spanning…

A perfect matching in a 4-uniform hypergraph is a subset of $\lfloor\frac{n}{4}\rfloor$ disjoint edges. We prove that if $H$ is a sufficiently large 4-uniform hypergraph on $n=4k$ vertices such that every vertex belongs to more than…

Given hypergraphs $F$ and $H$, an $F$-factor in $H$ is a set of vertex-disjoint copies of $F$ which cover all the vertices in $H$. Let $K^- _4$ denote the $3$-uniform hypergraph with $4$ vertices and $3$ edges. We show that for sufficiently…

We show that every 4-uniform hypergraph with $n$ vertices and minimum pair degree at least $(5/9+o(1))n^2/2$ contains a tight Hamiltonian cycle. This degree condition is asymptotically optimal.

We prove that for fixed $r\ge k\ge 2$, every $k$-uniform hypergraph on $n$ vertices having minimum codegree at least $(1-(\binom{r-1}{k-1}+\binom{r-2}{k-2})^{-1})n+o(n)$ contains the $(r-k+1)$th power of a tight Hamilton cycle. This result…

Given hypergraphs H and F, an F-factor in H is a spanning subgraph consisting of vertex disjoint copies of F. Let K_4^3-e denote the 3-uniform hypergraph on 4 vertices with 3 edges. We show that for \gamma>0 there exists an integer n_0 such…

Let $C_6^3$ be the 3-uniform hypergraph on $\{1,\dots, 6\}$ with edges $123, 345,561$, which can be seen as the triangle in 3-uniform hypergraphs. For sufficiently large $n$ divisible by 6, we show that every $n$-vertex 3-uniform hypergraph…

A result of Gy\'arf\'as says that for every $3$-coloring of the edges of the complete graph $K_n$, there is a monochromatic component of order at least $\frac{n}{2}$, and this is best possible when $4$ divides $n$. Furthermore, for all…

We establish an upper bound on the minimum codegree necessary for the existence of spanning, fractional Steiner triple systems in $3$-uniform hypergraphs. This improves upon a result by Lee in 2023. In particular, together with results from…

In this article we conjecture a 4-dimensional characterization of tightness: a contact structure is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces in Yx[0,1]. An affirmative answer to our conjecture…

Given an $r$-uniform hypergraph $H=(V,E)$ and a weight function $\omega:E\to\{1,\dots,w\}$, a coloring of vertices of $H$, induced by $\omega$, is defined by $c(v) = \sum_{e\ni v} w(e)$ for all $v\in V$. If there exists such a coloring that…

We prove that every (6k + 2l, 2k)-connected simple graph contains k rigid and l connected edge-disjoint spanning subgraphs. This implies a theorem of Jackson and Jord\'an [4] and a theorem of Jord\'an [6] on packing of rigid spanning…

Generalizing well-known results of Erd\H{o}s and Lov\'asz, we show that every graph $G$ contains a spanning $k$-partite subgraph $H$ with $\lambda{}(H)\geq \lceil{}\frac{k-1}{k}\lambda{}(G)\rceil$, where $\lambda{}(G)$ is the…

We study sufficient conditions for the existence of Hamilton cycles in uniformly dense $3$-uniform hypergraphs. Problems of this type were first considered by Lenz, Mubayi, and Mycroft for loose Hamilton cycles and Aigner-Horev and Levy…

We prove that every connected graph with $s$ vertices of degree~1 and 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${1\over 3}t +{1\over 4}s+{3\over 2}$ leaves. We present infinite series of graphs showing that…

We first prove that for every vertex x of a 4-connected graph G there exists a subgraph H in G isomorphic to a subdivision of the complete graph K4 on four vertices such that G-V(H) is connected and contains x. This implies an affirmative…