Related papers: Complete r-partite subgraphs of dense r-graphs
Let ${\cal{F}}=\{F_1,F_2,\ldots\}$ be a sequence of graphs such that $F_n$ is a graph on $n$ vertices with maximum degree at most $\Delta$. We show that there exists an absolute constant $C$ such that the vertices of any 2-edge-colored…
A central problem in extremal graph theory is to estimate, for a given graph $H$, the number of $H$-free graphs on a given set of $n$ vertices. In the case when $H$ is not bipartite, fairly precise estimates on this number are known. In…
A well-known theorem of Erd\H{o}s and Gallai asserts that a graph with no path of length $k$ contains at most $\frac{1}{2}(k-1)n$ edges. Recently Gy\H{o}ri, Katona and Lemons gave an extension of this result to hypergraphs by determining…
Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover number is at most $(r-1)\nu(H)$. This far reaching generalization of K\"onig's theorem is only known to be true for $r\leq 3$,…
Uncover the vertices of a given graph, deterministic or random, in random order; we consider both a discrete-time and a continuous-time version. We study the evolution of the number of visible edges, and show convergence after normalization…
Determine the size of $r$-graphs with given graph parameters is an interesting problem. Chv\'atal and Hanson (JCTB, 1976) gave a tight upper bound of the size of 2-graphs with restricted maximum degree and matching number; Khare (DM, 2014)…
Mixed graphs can be seen as digraphs with arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs,…
The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results…
In this paper we determine all the bipartite graphs with the maximum sum of squares of degrees among the ones with a given number of vertices and edges.
We study a variant of the Erd\H{o}s Matching Problem in random hypergraphs. Let $\mathcal{K}_p(n,k)$ denote the Erd\H{o}s-R\'enyi random $k$-uniform hypergraph on $n$ vertices where each possible edge is included with probability $p$. We…
We study the problem of recognizing graph powers and computing roots of graphs. We provide a polynomial time recognition algorithm for r-th powers of graphs of girth at least 2r+3, thus improving a bound conjectured by Farzad et al. (STACS…
The classical Zarankiewicz problem, which concerns the maximum number of edges in a bipartite graph without a forbidden complete bipartite subgraph, motivates a direct analogue for hypergraphs. Let $K_{s_1,\ldots, s_r}$ be the complete…
An $r$-uniform \textit{linear cycle} of length $\ell$, denoted by $C_{\ell}^r$, is an $r$-graph with edges $e_1, \ldots, e_{\ell}$ such that for every $i\in [\ell-1]$, $|e_i\cap e_{i+1}|=1$, $|e_{\ell}\cap e_1|=1$ and $e_i\cap…
Motivated by recent extensive studies on Wenger graphs, we introduce a new infinite class of bipartite graphs of the similar type, called linearized Wenger graphs. The spectrum, diameter and girth of these linearized Wenger graphs are…
For a graph G and integer r\geq 1 we denote the collection of independent r-sets of G by I^{(r)}(G). If v\in V(G) then I_v^{(r)}(G) is the collection of all independent r-sets containing v. A graph G, is said to be r-EKR, for r\geq 1, iff…
In a simple drawing of a graph every pair of edges intersect each other in at most one point, which is either a common endvertex or a proper crossing. For each positive integer $n$, Negami identified a drawing $B_n$ of the complete…
We show that there is a constant c>0 so that for any fixed r which is at least 3 a.a.s. an r-regular graph on n vertices contains a complete graph on c n^{1/2} vertices as a minor. This confirms a conjecture of Markstrom. Since any minor of…
We prove that the number of vertices of a polytope of a particular kind is exponentially large in the dimension of the polytope. As a corollary, we prove that an n-dimensional centrally symmetric polytope with O(n) facets has 2^{Omega(n)}…
For given graphs G1 and G2 the Ramsey number R(G1,G2), is the smallest positive integer n such that each blue-red edge coloring of the complete graph Kn contains a blue copy of G1 or a red copy of G2. In 1983, Erdos conjectured that there…
In 1975 Lov\'{a}sz conjectured that every $r$-partite, $r$-uniform hypergraph contains $r-1$ vertices whose deletion reduces the matching number. If true, this statement would imply a well-known conjecture of Ryser from 1971, which states…