Related papers: Spanning spheres in Dirac hypergraphs

Hypergraphs, as a generalization of simplicial complexes, have long been a subject of interest in their geometric interpretation. The subdivision of simplicial complexes can, to some extent, provide insights into the geometry of simplicial…

A subset $C$ of edges in a $k$-uniform hypergraph $H$ is a \emph{loose Hamilton cycle} if $C$ covers all the vertices of $H$ and there exists a cyclic ordering of these vertices such that the edges in $C$ are segments of that order and such…

Let $\Delta$ denote a non-degenerate $k$-simplex in $\mathbb{R}^k$. The set $\text{Sim}(\Delta)$ of simplices in $\mathbb{R}^k$ similar to $\Delta$ is diffeomorphic to $O(k)\times [0,\infty)\times \mathbb{R}^k$, where the factor in $O(k)$…

We consider when automorphisms of a graph can be induced by homeomorphisms of embeddings of the graph in a $3$-manifold. In particular, we prove that every automorphism of a graph is induced by a homeomorphism of some embedding of the graph…

To a rational homology sphere graph manifold one can associate a weighted tree invariant called splice diagram. In this article we prove a sufficient numerical condition on the splice diagram for a graph manifold to be a singularity link.…

The classical Dirac theorem asserts that every graph $G$ on $n$ vertices with minimum degree $\delta(G) \ge \lceil n/2 \rceil$ is Hamiltonian. The lower bound of $\lceil n/2 \rceil$ on the minimum degree of a graph is tight. In this paper,…

In a seminal work, K\"uhn, Osthus, Townsend, and Zhao used the hypergraph container method to determine the typical structure of oriented graphs and digraphs avoiding a fixed tournament or cycle. Their main tool, a container theorem for…

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…

An Euler tour in a hypergraph $H$ is a closed walk that traverses each edge of $H$ exactly once, and an Euler family is a family of closed walks that jointly traverse each edge of $H$ exactly once. An $\ell$-covering $k$-hypergraph, for $2…

In the distributed subgraph-freeness problem, we are given a graph $H$, and asked to determine whether the network graph contains $H$ as a subgraph or not. Subgraph-freeness is an extremely local problem: if the network had no bandwidth…

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…

Let S be a compact connected surface and let f be an element of the group Homeo\_0(S) of homeomorphisms of S isotopic to the identity. Denote by \tilde{f} a lift of f to the universal cover of S. Fix a fundamental domain D of this universal…

Let $H=(V,E)$ be a hypergraph, where $V$ is a set of vertices and $E$ is a set of non-empty subsets of $V$ called edges. If all edges of $H$ have the same cardinality $r$, then $H$ is a $r$-uniform hypergraph; if $E$ consists of all…

Given a tree T on v vertices and an integer k exceeding one. One can define the k-expansion T^k as a k-uniform linear hypergraph by enlarging each edge with a new, distinct set of (k-2) vertices. Then T^k has v+ (v-1)(k-2) vertices. The aim…

In this paper we investigate Ramanujan hypergraphs by using hypergraph coverings. We first show that the spectrum of a $k$-fold covering $\bar{H}$ of a connected hypergraph $H$ contains the spectrum of $H$, and that it is the union of the…

Given a graph $G$, we say a $k$-uniform hypergraph $H$ on the same vertex set contains a Berge-$G$ if there exists an injection $\phi:E(G)\to E(H)$ such that $e\subseteq\phi(e)$ for each edge $e\in E(G)$. A hypergraph $H$ is…

A classic theorem of Dirac from 1952 states that every graph with minimum degree at least n/2 contains a Hamiltonian cycle. In 1963, P\'osa conjectured that every graph with minimum degree at least 2n/3 contains the square of a Hamiltonian…

We prove the well-known Brown-Erd\H{o}s-S\'os Conjecture for hypergraphs of large uniformity in the following form: any dense linear $r$-graph $G$ has $k$ edges spanning at most $(r-2)k+3$ vertices, provided the uniformity $r$ of $G$ is…

Many complex systems show non-pairwise interactions, which can be captured by hypergraphs. In this work, we propose an edge-swapping method to sample random directed hypergraphs with fixed vertex and hyperarc degrees, which can be applied…

We consider the problem of finding small distance-preserving subgraphs of undirected, unweighted interval graphs with $k$ terminal vertices. To start with, we show that finding an optimal distance-preserving subgraph is $\mathsf{NP}$-hard…