Related papers: Cycles in Random Bipartite Graphs

Let $G$ be any graph of minimum degree at least $k$, and let $G_p$ be the random subgraph of $G$ obtained by keeping each edge independently with probability $p$. Recently, Krivelevich, Lee and Sudakov showed that if $pk\to\infty$ then with…

For a graph $G$ and $p\in [0,1]$, let $G_p$ arise from $G$ by deleting every edge mutually independently with probability $1-p$. The random graph model $(K_n)_p$ is certainly the most investigated random graph model and also known as the…

Magnant and Martin conjectured that the vertex set of any $d$-regular graph $G$ on $n$ vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this…

We consider the problem of partitioning the edge set of a graph $G$ into the minimum number $\tau(G)$ of edge-disjoint complete bipartite subgraphs. We show that for a random graph $G$ in $G(n,p)$, for $p$ is a constant no greater than…

Let $G$ be a 2-connected $n$-vertex graph and $N_s(G)$ be the total number of $s$-cliques in $G$. Let $k\ge 4$ and $s\ge 2$ be integers. In this paper, we show that if $G$ has an edge $e$ which is not on any cycle of length at least $k$,…

We study graphs on $n$ vertices which have $2n-2$ edges and no proper induced subgraphs of minimum degree $3$. Erd\H{o}s, Faudree, Gy\'arf\'as, and Schelp conjectured that such graphs always have cycles of lengths $3,4,5,\dots, C(n)$ for…

A graph $G$ is called $C_{2k+1}$-free if it does not contain any cycle of length $2k+1$. In 1981, Haggkvist, Faudree and Schelp showed that every $n$-vertex triangle-free graph with more than $\frac{(n-1)^2}{4}+1$ edges is bipartite. In…

For graphs $G_0$, $G_1$ and $G_2$, write $G_0\longmapsto(G_1, G_2)$ if each red-blue-edge-coloring of $G_0$ yields a red $G_1$ or a blue $G_2$. The Ramsey number $r(G_1, G_2)$ is the minimum number $n$ such that the complete graph…

We solve four similar problems: For every fixed $s$ and large $n$, we describe all values of $n_1,\ldots,n_s$ such that for every $2$-edge-coloring of the complete $s$-partite graph $K_{n_1,\ldots,n_s}$ there exists a monochromatic (i)…

The binomial random bipartite graph $G(n,n,p)$ is the random graph formed by taking two partition classes of size $n$ and including each edge between them independently with probability $p$. It is known that this model exhibits a similar…

A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We show that every planar graph without cycles of length 4 or 5 is…

We prove that a strongly connected balanced bipartite directed graph of order $2a\geq 6$ with partite sets $X$ and $Y$ contains cycles of every length $2, 4, \ldots , 2a$, provided $d(x)+d(y)\geq 3a$ for every pair of vertices $x$, $y$…

Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $\{1, \ldots, n\}$ with $m=m(n)$ edges. We study the cycle and block structure of $P(n,m)$ when $m\sim n/2$. More precisely, we determine…

For fixed $k\ge 2$, determining the order of magnitude of the number of edges in an $n$-vertex bipartite graph not containing $C_{2k}$, the cycle of length $2k$, is a long-standing open problem. We consider an extension of this problem to…

Recently, working on the Tanner graph which represents a low density parity check (LDPC) code becomes an interesting research subject. Finding the number of short cycles of Tanner graphs motivated Blake and Lin to investigate the…

In a graph $G$, let $\mu_G(xy)$ denote the number of edges between $x$ and $y$ in $G$. Let $\lambda K_{v,u}$ be the graph $(V\cup U,E)$ with $|V|=v$, $|U|=u$, and \[ \mu_G(xy)=\begin{cases} \lambda &\mbox{if $x\in U$ and $y\in V$ or if…

For a given graph $G$ of minimum degree at least $k$, let $G_p$ denote the random spanning subgraph of $G$ obtained by retaining each edge independently with probability $p=p(k)$. We prove that if $p \ge \frac{\log k + \log \log k +…

Recently there has been much interest in studying random graph analogues of well known classical results in extremal graph theory. Here we follow this trend and investigate the structure of triangle-free subgraphs of $G(n,p)$ with high…

In his seminal 1976 paper, P\'osa showed that for all $p\geq C\log n/n$, the binomial random graph $G(n,p)$ is with high probability Hamiltonian. This leads to the following natural questions, which have been extensively studied: How well…

Given a graph $G$ and a subset $X$ of vertices of $G$ with size at least two, we denote by $N^2_G(X)$ the set of vertices of $G$ that have at least two neighbors in $X$. We say that a bipartite graph $G$ with sides $A$ and $B$ satisfies the…