Related papers: Maximum star densities

For any countably infinite graph $G$, Ramsey's theorem guarantees an infinite monochromatic copy of $G$ in any $r$-coloring of the edges of the countably infinite complete graph $K_\mathbb{N}$. Taking this a step further, it is natural to…

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…

A graph is called $k$-critical if its chromatic number is $k$ but any proper subgraph has chromatic number less than $k$. An old and important problem in graph theory asks to determine the maximum number of edges in an $n$-vertex…

For a fixed infinite graph $H$, we study the largest density of a monochromatic subgraph isomorphic to $H$ that can be found in every two-coloring of the edges of $K_{\mathbb{N}}$. This is called the Ramsey upper density of $H$, and was…

In a graph $G$ of maximum degree 3, let $\gamma(G)$ denote the largest fraction of edges that can be 3 edge-coloured. Rizzi \cite{Riz09} showed that $\gamma(G) \geq 1-\frac{2\strut}{\strut 3 g_{odd}(G)}$ where $g_{odd}(G)$ is the odd girth…

Bollob\'as and Nikiforov conjectured that for any graph $G \neq K_n$ with $m$ edges \[ \lambda_1^2+\lambda_2^2\le \bigg( 1-\frac{1}{\omega(G)}\bigg)2m\] where $\lambda_1$ and $\lambda_2$ denote the two largest eigenvalues of the adjacency…

We study a model of random uniform hypergraphs, where a random instance is obtained by adding random edges to a large hypergraph of a given density. We obtain a tight bound on the number of random edges required to ensure…

We present spectra of the most massive quiescent galaxy yet discovered at $z>3$, spectroscopically confirmed via the detection of Balmer absorption features in the $H-$ and $K-$bands of Keck/MOSFIRE. The spectra confirm a galaxy with no…

A clique covering of a graph $G$ is a set of cliques of $G$ such that any edge of $G$ is contained in one of these cliques, and the weight of a clique covering is the sum of the sizes of the cliques in it. The sigma clique cover number…

We study how many copies of a graph $F$ that another graph $G$ with a given number of cliques is guaranteed to have. For example, one of our main results states that for all $t\ge 2$, if $G$ is an $n$ vertex graph with $kn^{3/2}$ triangles…

For $q>27$ we determine the independence number $\alpha(\Gamma)$ of the Kneser graph $\Gamma$ on plane-solid flags in $PG(6,q)$. More precisely we describe all maximal independent sets of size at least $q^{11}$ and show that every other…

We investigate the effect of a fixed forbidden clique minor upon the strong chromatic index, both in multigraphs and in simple graphs. We conjecture for each $k\ge 4$ that any $K_k$-minor-free multigraph of maximum degree $\Delta$ has…

This paper proposes k-medoids clustering method to reveal the distinct groups of 1,318 variable stars in the Galaxy based on their light curves, where each light curve represents the graph of brightness of the star against time. To overcome…

Let $r\ge 3$. Given an $r$-graph $H$, the minimum codegree $\delta_{r-1}(H)$ is the largest integer $t$ such that every $(r-1)$-subset of $V(H)$ is contained in at least $t$ edges of $H$. Given an $r$-graph $F$, the codegree Tur\'an density…

In this paper we disprove a conjecture of Lidick\'y and Murphy about the number of copies of a given graph in a $K_r$-free graph and give an alternative general conjecture. We also prove an asymptotically tight bound on the number of copies…

Given a graph $G$ and a positive integer $k$, the \emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) copy of $G$ or a…

A set $D$ of vertices of a graph $G$ is a dissociation set if each vertex of $D$ has at most one neighbor in $D$. The dissociation number of $G$, $diss(G)$, is the cardinality of a maximum dissociation set in a graph $G$. In this paper we…

We prove that for every non-trivial hereditary family of graphs ${\cal P}$ and for every fixed $p \in (0,1)$, the maximum possible number of edges in a subgraph of the random graph $G(n,p)$ which belongs to ${\cal P}$ is, with high…

A bound on consecutive clique numbers of graphs is established. This bound is evaluated and shown to often be much better than the bound of the Kruskal-Katona theorem. A bound on non-consecutive clique numbers is also proven.

The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In: Towards a…