Related papers: The genus of a random bipartite graph

For a graph $G$, denote by $t_r(G)$ (resp. $b_r(G)$) the maximum size of a $K_r$-free (resp. $(r-1)$-partite) subgraph of $G$. Of course $t_r(G) \geq b_r(G)$ for any $G$, and Tur\'an's Theorem says that equality holds for complete graphs.…

We investigate the genus $g(n,m)$ of the Erd\H{o}s-R\'enyi random graph $G(n,m)$, providing a thorough description of how this relates to the function $m=m(n)$, and finding that there is different behaviour depending on which `region' $m$…

Let $G$ be a finite graph with minimum degree $r$. Form a random subgraph $G_p$ of $G$ by taking each edge of $G$ into $G_p$ independently and with probability $p$. We prove that for any constant $\epsilon>0$, if $p=\frac{1+\epsilon}{r}$,…

We provide a sufficient condition on the isoperimetric properties of a regular graph $G$ of growing degree $d$, under which the random subgraph $G_p$ typically undergoes a phase transition around $p=\frac{1}{d}$ which resembles the…

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…

In the classical Erd\"os-R\'enyi random graph G(n,p) there are n vertices and each of the possible edges is independently present with probability p. The random graph G(n,p) is homogeneous in the sense that all vertices have the same…

A graph is even-degenerate if one can iteratively remove a vertex of even degree at each step until at most one edge remains. Recently, Janzer and Yip showed that the Erd\H{o}s--Renyi random graph $G(n,1/2)$ is even-degenerate with high…

Consider the uniform random graph $G(n,M)$ with $n$ vertices and $M$ edges. Erd\H{o}s and R\'enyi (1960) conjectured that the limit $$ \lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} $$ exists and is a constant strictly…

In 1980, Ajtai, Komlos and Szemer{\'e}di defined "groupie": Let $G=(V,E)$ be a simple graph, $|V|=n$, $|E|=e$. For a vertex $v\in V$, let $r(v)$ denote the sum of the degrees of the vertices adjacent to $v$. We say $v\in V$ is a {\it…

We study a random graph model which combines properties of the edge percolation model on Z^d and a classical random graph G(n,c/n). We show that this model, being a homogeneous random graph, has a natural relation to the so-called "rank 1…

Let $G_{n,p}$ be the standard Erd\H{o}s-R\'enyi-Gilbert random graph and let $G_{n,n,p}$ be the random bipartite graph on $n+n$ vertices, where each $e\in [n]^2$ appears as an edge independently with probability $p$. For a graph $G=(V,E)$,…

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…

We study a random graph model which is a superposition of the bond percolation model on $Z^d$ with probability $p$ of an edge, and a classical random graph $G(n, c/n)$. We show that this model, being a {\it homogeneous} random graph, has a…

Random graphs with latent geometric structure are popular models of social and biological networks, with applications ranging from network user profiling to circuit design. These graphs are also of purely theoretical interest within…

A graph $G$ on $n$ vertices is \textit{pancyclic} if it contains cycles of length $t$ for all $3 \leq t \leq n$. In this paper we prove that for any fixed $\epsilon>0$, the random graph $G(n,p)$ with $p(n)\gg n^{-1/2}$ asymptotically almost…

We consider the spectral gap of a uniformly chosen random $(d_1,d_2)$-biregular bipartite graph $G$ with $|V_1|=n, |V_2|=m$, where $d_1,d_2$ could possibly grow with $n$ and $m$. Let $A$ be the adjacency matrix of $G$. Under the assumption…

We report on the phase transition of finding a complete subgraph, of specified dimensions, in a bipartite graph. Finding a complete subgraph in a bipartite graph is a problem that has growing attention in several domains, including…

Settling a first case of a conjecture of M. Kahle on the homology of the clique complex of the random graph $G=G_{n,p}$, we show, roughly speaking, that (with high probability) the triangles of $G$ span its cycle space whenever each of its…

A simple undirected graph is said to be {\em semisymmetric} if it is regular and edge-transitive but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two parts of equal size. It was proved in [{\em J. Combin.…

We prove that if $p\ge n^{-\frac{1}{3}+\beta}$ for some $\beta > 0$, then asymptotically almost surely the binomial random graph $G(n,p)$ has a $K_3$-packing containing all but at most $n + O(1)$ edges. Similarly, we prove that if $d \ge…