Related papers: Geometric Progression-Free Sequences with Small Ga…

Rank-width of a graph G, denoted by rw(G), is a width parameter of graphs introduced by Oum and Seymour (2006). We investigate the asymptotic behavior of rank-width of a random graph G(n,p). We show that, asymptotically almost surely, (i)…

Two central topics of study in combinatorics are the so-called evolution of random graphs, introduced by the seminal work of Erd\H{o}s and R\'enyi, and the family of $H$-free graphs, that is, graphs which do not contain a subgraph…

Given a graph $G(V,E)$, a vertex subset $S$ of $G$ is called an open packing in $G$ if no pair of distinct vertices in $S$ have a common neighbour in $G$. The size of a largest open packing in $G$ is called the open packing number,…

We establish explicit unconditional results on the graphic properties of the prime gap sequence. Let $p_n$ denote the $n$-th prime number (with $p_0=1$) and $\mathrm{PD}_n = (p_\ell - p_{\ell-1})_{\ell=1}^n$ be the sequence of the first $n$…

In their 1997 paper titled ``Fruit Salad", Gy\'{a}rf\'{a}s posed the following conjecture: there exists a constant $k$ such that if each path of a graph spans a $3$-colourable subgraph, then the graph is $k$-colourable. It is noted that…

Since planar triangle-free graphs are 3-colourable, such a graph with n vertices has an independent set of size at least n/3. We prove that unless the graph contains a certain obstruction, its independence number is at least n/(3-epsilon)…

Pach and Palincza proved the following generalization of Ellenberg and Gijswijt's bound for the size of $k$-term arithmetic progression-free subsets, where $k\in \{4,5,6\}$: Let $m>0$ be an integer such that $6$ divides $m$ and let $k\in…

We give a method of generating strongly polynomial sequences of graphs, i.e., sequences $(H_{\mathbf{k}})$ indexed by a multivariate parameter $\mathbf{k}=(k_1,\ldots, k_h)$ such that, for each fixed graph $G$, there is a multivariate…

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. A graph $G$ is $k$-vertex-critical if every proper induced subgraph of $G$ has chromatic number less than $k$,…

A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic…

Consider the triangle-free graph process, which starts from the empty graph on $n$ vertices and a random ordering of the possible ${n \choose 2}$ edges; the edges are added in this ordering provided the graph remains triangle free. We will…

Question 10208b (1992) of the American Mathematical Monthly asked: does there exist an increasing sequence $\{a_k\}$ of positive integers and a constant $B > 0$ having the property that $\{ a_k + n\}$ contains no more than $B$ primes for…

An integer sequence is said to be 3-free if no three elements form an arithmetic progression. Following the greedy algorithm, the Stanley sequence $S(a_0,a_1,\ldots,a_k)$ is defined to be the 3-free sequence $\{a_n\}$ having initial terms…

We show that for some constant $\beta > 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^\beta)} \cdot N$ contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic…

We show that for any constant $\mu>0$ and $k\ge 3$, there exists $\alpha>0$ such that the following holds for sufficiently large $n \in \mathbb{N}$. If $G=(V_{1},\ldots,V_{k},E)$ is a spanning subgraph of the $n$-blow-up of $K_{k}$ with…

For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. In this paper, we study the problem of bounding t(G) for graphs which do not contain a complete graph K_r on r vertices. This problem…

We present a variety of numerical data related to the growth of terms in aliquot sequences, iterations of the function $s(n) = \sigma(n) - n$. First, we compute the geometric mean of the ratio $s_k(n)/s_{k-1}(n)$ of $k$th iterates for $n…

Let $\alpha=\alpha(G)$ be the independence number of a simple graph $G$ with $n$ vertices and $I(G)$ be its edge ideal in $S=K[x_1,\ldots, x_n]$. If $S/I(G)$ is Gorenstein, the graph $G$ is called Gorenstein over $K$ and if $G$ is…

An $(\alpha,\beta)$-spanner of an $n$-vertex graph $G=(V,E)$ is a subgraph $H$ of $G$ satisfying that $dist(u, v, H) \leq \alpha \cdot dist(u, v, G)+\beta$ for every pair $(u, v)\in V \times V$, where $dist(u,v,G')$ denotes the distance…

In a graph whose vertices are assigned integer ranks, a path is well-ranked if the endpoints have distinct ranks or some interior point has a higher rank than the endpoints. A ranking is an assignment of ranks such that all nontrivial paths…