Related papers: On regular hypergraphs of high girth

We investigate the occurrence of powers of tight Hamilton cycles in random hypergraphs. For every $r\ge 3$ and $k\ge 1$, we show that there exists a constant $C > 0$ such that if $p=p(n) \ge Cn^{-1/\binom{k+r-2}{r-1}}$ then asymptotically…

This paper focuses on extensions of the classic Erd\H{o}s-Gallai Theorem for the set of weighted function of each edge in a graph. The weighted function of an edge $e$ of an $n$-vertex uniform hypergraph $\mathcal{H}$ is defined to a…

We estimate the likely values of the chromatic and independence numbers of the random $r$-uniform $d$-regular hypergraph on $n$ vertices for fixed $r$, large fixed $d$, and $n \rightarrow \infty$.

A uniformly random graph on $n$ vertices with a fixed degree sequence, obeying a $\gamma$ subpower law, is studied. It is shown that, for $\gamma>3$, in a subcritical phase with high probability the largest component size does not exceed…

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…

In this paper, we study the maximum number of edges in an $N$-vertex $r$-uniform hypergraph with girth $g$ where $g \in \{5,6 \}$. Writing $\textrm{ex}_r ( N, \mathcal{C}_{<g} )$ for this maximum, it is shown that $\textrm{ex}_r ( N ,…

We study the emergence of loose Hamilton cycles in subgraphs of random hypergraphs. Our main result states that the minimum $d$-degree threshold for loose Hamiltonicity relative to the random $k$-uniform hypergraph $H_k(n,p)$ coincides with…

What is the probability that a sparse $n$-vertex random $d$-regular graph $G_n^d$, $n^{1-c}<d=o(n)$ contains many more copies of a fixed graph $K$ than expected? We determine the behavior of this upper tail to within a logarithmic gap in…

We consider the geometric random (GR) graph on the $d-$dimensional torus with the $L_\sigma$ distance measure ($1 \leq \sigma \leq \infty$). Our main result is an exact characterization of the probability that a particular labeled cycle…

We generalize the fact that graphs with small VC-dimension can be approximated by rectangles, showing that hypergraphs with small VC_k-dimension (equivalently, omitting a fixed finite (k+1)-partite (k+1)-uniform hypergraph) can be…

Dirac proved that each $n$-vertex $2$-connected graph with minimum degree at least $k$ contains a cycle of length at least $\min\{2k, n\}$. We consider a hypergraph version of this result. A Berge cycle in a hypergraph is an alternating…

Consider a graph $G=(V,E)$ without isolated edges and with maximum degree $\Delta$. Given a colouring $c:E\to\{1,2,\ldots,k\}$, the weighted degree of a vertex $v\in V$ is the sum of its incident colours, i.e., $\sum_{e\ni v}c(e)$. For any…

We consider $p$-orientations, which are defined to be orientations of $d$-regular graphs such that every vertex either has in-degree $p$ or out-degree $p$. These generalise the orientations considered in Jaeger's conjecture, where $d=4p+1$.…

In this paper we show that $e/n$ is the sharp threshold for the existence of tight Hamilton cycles in random $k$-uniform hypergraphs, for all $k\ge 4$. When $k=3$ we show that $1/n$ is an asymptotic threshold. We also determine thresholds…

Let $\mathcal{G}(n,r,s)$ denote a uniformly random $r$-regular $s$-uniform hypergraph on $n$ vertices, where $s$ is a fixed constant and $r=r(n)$ may grow with $n$. An $\ell$-overlapping Hamilton cycle is a Hamilton cycle in which…

We prove new lower bounds on the modularity of graphs. Specifically, the modularity of a graph $G$ with average degree $\bar d$ is $\Omega(\bar{d}^{-1/2})$, under some mild assumptions on the degree sequence of $G$. The lower bound…

The maximum number of vertices in a graph of maximum degree $\Delta\ge 3$ and fixed diameter $k\ge 2$ is upper bounded by $(1+o(1))(\Delta-1)^{k}$. If we restrict our graphs to certain classes, better upper bounds are known. For instance,…

Given a group $G$, the model $\mathcal{G}(G,p)$ denotes the probability space of all Cayley graphs of $G$ where each element of $G$ is included in the generating set independently at random with probability $p$. In this article, we…

The Moore bound constitutes both an upper bound on the order of a graph of maximum degree $d$ and diameter $D=k$ and a lower bound on the order of a graph of minimum degree $d$ and odd girth $g=2k+1$. Graphs missing or exceeding the Moore…

We prove upper bounds on the face numbers of simplicial complexes in terms on their girths, in analogy with the Moore bound from graph theory. Our definition of girth generalizes the usual definition for graphs.