Related papers: Clique factors in pseudorandom graphs

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…

We study robust versions of properties of $(n,d,\lambda)$-graphs, namely, the property of a random sparsification of an $(n,d,\lambda)$-graph, where each edge is retained with probability $p$ independently. We prove such results for the…

To any simple graph $G$, the clique graph operator $K$ associates the graph $K(G)$ which is the intersection graph of the maximal complete subgraphs of $G$. The iterated clique graphs are defined by $K^{0}(G)=G$ and $K^{n}(G)=K(K^{n-1}(G))$…

The theory of dense graph limits comes with a natural sampling process which yields an inhomogeneous variant G(n,W) of the Erdos-Renyi random graph. Here we study the clique number of these random graphs. We establish the concentration of…

A sparse version of Mantel's Theorem is that, for sufficiently large $p$, with high probability (w.h.p.), every maximum triangle-free subgraph of $G(n,p)$ is bipartite. DeMarco and Kahn proved this for $p>K \sqrt{\log n/n}$ for some…

Let $G$ be a connected simple graph of order $n$ and let $\Delta(G)$ and $\chi'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $\chi'(G)=\Delta(G)$ or $\Delta(G)+1$. Following this result, $G$ is…

A graph is prime (with respect to the split decomposition) if its vertex set does not admit a partition (A,B) (called a split) with |A|, |B| >= 2 such that the set of edges joining A and B induces a complete bipartite graph. We prove that…

A good edge-labelling of a simple graph is a labelling of its edges with real numbers such that, for any ordered pair of vertices (u,v), there is at most one nondecreasing path from u to v. Say a graph is good if it admits a good…

In a graph $G$, a subset of vertices $S \subseteq V(G)$ is said to be cyclable if there is a cycle containing the vertices in some order. $G$ is said to be $k$-cyclable if any subset of $k \geq 2$ vertices is cyclable. If any $k$…

Tur\'{a}n's theorem is a cornerstone of extremal graph theory. It asserts that for any integer $r \geq 2$ every graph on $n$ vertices with more than ${\tfrac{r-2}{2(r-1)}\cdot n^2}$ edges contains a clique of size $r$, i.e., $r$ mutually…

A graph is called an $(r,k)$-graph if its vertex set can be partitioned into $r$ parts, each having at most $k$ vertices and there is at least one edge between any two parts. Let $f(r,H)$ be the minimum $k$ for which there exists an…

A construction of Alon and Krivelevich gives highly pseudorandom $K_k$-free graphs on $n$ vertices with edge density equal to $\Theta(n^{-1/(k -2)})$. In this short note we improve their result by constructing an infinite family of highly…

For integers $k \geq 3$ and $r\geq 2$, we show that for every $\alpha> 0$, there exists $\varepsilon > 0$ such that the union of $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $\alpha n$ and a binomial random…

The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and…

The toughness of a graph $G$ is defined as the minimum value of $|S|/c(G-S)$ over all cutsets $S$ of $G$ if $G$ is noncomplete, and is defined to be $\infty$ if $G$ is complete. For a real number $t$, we say that $G$ is $t$-tough if its…

We prove that any $n$-vertex graph whose complement is triangle-free contains $n^2/12-o(n^2)$ edge-disjoint triangles. This is tight for the disjoint union of two cliques of order $n/2$. We also prove a corresponding stability theorem, that…

A graph $G = (V, E)$ is \emph{partitionable} if there exists a partition $\{A, B\}$ of $V$ such that $A$ induces a disjoint union of cliques and $B$ induces a triangle-free graph. In this paper we investigate the computational complexity of…

Let \mathcal{F}_k denote the family of 2-edge-colored complete graphs on 2k vertices in which one color forms either a clique of order k or two disjoint cliques of order k. Bollob\'as conjectured that for every \epsilon>0 and positive…

Let $k\geq 2$. We show that, for a sufficiently small $\varepsilon>0$, any sufficiently large $n$-vertex Hamiltonian graph of minimum degree at least $n^{1-\varepsilon}$ contains a $2$-factor consisting of exactly $k$ cycles. This is the…

B. Bollob\'{a}s and G. Brightwell and independently R. Shi proved the existence of a cycle through all vertices whose degrees at least $\frac{n}{2}$ in any $2$-connected graph of order $n$. Motivated by this result, we prove the existence…