Related papers: Spanning universality in random graphs

A graph $G$ is said to be $\mathcal H(n,\Delta)$-universal if it contains every graph on $n$ vertices with maximum degree at most $\Delta$. It is known that for any $\varepsilon > 0$ and any natural number $\Delta$ there exists $c > 0$ such…

In this paper the problem of finding various spanning structures in random hypergraphs is studied. We notice that a general result of Riordan [Spanning subgraphs of random graphs, Combinatorics, Probability & Computing 9 (2000), no. 2,…

For a family $\mathcal{F}$ of graphs, a graph $G$ is called \emph{$\mathcal{F}$-universal} if $G$ contains every graph in $\mathcal{F}$ as a subgraph. Let $\mathcal{F}_n(d)$ be the family of all graphs on $n$ vertices with maximum degree at…

Given a class of graphs F, we say that a graph G is universal for F, or F-universal, if every H in F is contained in G as a subgraph. The construction of sparse universal graphs for various families F has received a considerable amount of…

In this paper we show how to use simple partitioning lemmas in order to embed spanning graphs in a typical member of $G(n,p)$. Let the \emph{maximum density} of a graph $H$ be the maximum average degree of all the subgraphs of $H$. First,…

In this paper we consider the problem of embedding almost-spanning, bounded degree graphs in a random graph. In particular, let $\Delta\geq 5$, $\varepsilon > 0$ and let $H$ be a graph on $(1-\varepsilon)n$ vertices and with maximum degree…

A graph $G$ is called universal for a family of graphs $\mathcal{F}$ if it contains every element $F \in \mathcal{F}$ as a subgraph. Let $\mathcal{F}(n,2)$ be the family of all graphs with maximum degree $2$. Ferber, Kronenberg, and Luh…

A tuple $(G_1,\dots,G_n)$ of graphs on the same vertex set of size $n$ is said to be Hamilton-universal if for every map $\chi: [n]\to[n]$ there exists a Hamilton cycle whose $i$-th edge comes from $G_{\chi(i)}$. Bowtell, Morris, Pehova and…

A graph $G$ is $\textit{universal}$ for a (finite) family $\mathcal{H}$ of graphs if every $H \in \mathcal{H}$ is a subgraph of $G$. For a given family $\mathcal{H}$, the goal is to determine the smallest number of edges an…

Given a family of hypergraphs $\mathcal{H}$, we say that a hypergraph $\Gamma$ is $\mathcal{H}$-universal if it contains every $H \in \mathcal{H}$ as a subgraph. For $D, r \in \mathbb{N}$, we construct an $r$-uniform hypergraph with…

For a family of graphs $\mathcal{F}$, a graph $G$ is $\mathcal{F}$-universal if $G$ contains every graph in $\mathcal{F}$ as a (not necessarily induced) subgraph. For the family of all graphs on $n$ vertices and of maximum degree at most…

We investigate the asymptotic structure of a random perfect graph $P_n$ sampled uniformly from the perfect graphs on vertex set $\{1,\ldots,n\}$. Our approach is based on the result of Pr\"omel and Steger that almost all perfect graphs are…

A hypergraph $H$ is called universal for a family $\mathcal{F}$ of hypergraphs, if it contains every hypergraph $F \in \mathcal{F}$ as a copy. For the family of $r$-uniform hypergraphs with maximum vertex degree bounded by $\Delta$ and at…

For graphs $G$ and $H$, let $G\to (H,H)$ signify that any red/blue edge coloring of $G$ contains a monochromatic $H$ as a subgraph, and $\mathcal{H}(\Delta,n)=\{H:|V(H)|=n,\Delta(H)\le \Delta\}$. For fixed $\Delta$ and $n$, we say that $G$…

We solve a problem of Krivelevich, Kwan and Sudakov [SIAM Journal on Discrete Mathematics 31 (2017), 155-171] concerning the threshold for the containment of all bounded degree spanning trees in the model of randomly perturbed dense graphs.…

Given a collection $\mathcal{G}=(G_1,\dots, G_h)$ of graphs on the same vertex set $V$ of size $n$, an $h$-edge graph $H$ on the vertex set $V$ is a $\mathcal{G}$-transversal if there exists a bijection $\lambda : E(H) \rightarrow [h]$ such…

Let $\Delta \ge 3$ be fixed, $n \ge n_\Delta$ be a large integer. It is a classical result that $\Delta$--regular expanders on $n$ vertices are not embeddable as geometric (distance) graphs into Euclidean space of dimension less than $c…

Let Delta>1 be a fixed integer. We show that the random graph G(n,p) with p>>(log n/n)^{1/Delta} is robust with respect to the containment of almost spanning bipartite graphs H with maximum degree Delta and sublinear bandwidth in the…

We study the model $G_\alpha\cup G(n,p)$ of randomly perturbed dense graphs, where $G_\alpha$ is any $n$-vertex graph with minimum degree at least $\alpha n$ and $G(n,p)$ is the binomial random graph. We introduce a general approach for…

We introduce a natural generalization of the Erd\H{o}s-R\'enyi random graph model in which random instances of a fixed motif are added independently. The binomial random motif graph $G(H,n,p)$ is the random (multi)graph obtained by adding…