Related papers: Kneser ranks of random graphs and minimum differen…

Generally, a graph G, an independent set is a subset S of vertices in G such that no two vertices in S are adjacent (connected by an edge) and a vertex cover is a subset S of vertices such that each edge of G has at least one of its…

A vertex cut $S$ of a connected graph $G$ is a subset of vertices of $G$ whose deletion makes $G$ disconnected. A super vertex cut $S$ of a connected graph $G$ is a subset of vertices of $G$ whose deletion makes $G$ disconnected and there…

Menger's theorem is an important building block of numerous results in the study of graph structure. We consider a variant in terms of coarse geometry. We say that a set of graphs has the weak coarse Menger property if there exist functions…

We discuss the topological properties of the independence complex of Kneser graphs, Ind(KG$(n, k))$, with $n\geq 3$ and $k\geq 1$. By identifying one kind of maximal simplices through projective planes, we obtain homology generators for the…

Let $G$ be a finite simple graph. For $X \subset V(G)$, the difference of $X$, $d(X) := |X| - |N (X)|$ where $N(X)$ is the neighborhood of $X$ and $\max \, \{d(X):X\subset V(G)\}$ is called the critical difference of $G$. $X$ is called a…

Let $\delta$ and $\Delta$ be the minimum and the maximum degree of the vertices of a simple connected graph $G$, respectively. The distinguishing index of a graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of…

For graphs $F$ and $H$, let $f_{F,H}(n)$ be the minimum possible size of a maximum $F$-free induced subgraph in an $n$-vertex $H$-free graph. This notion generalizes the Ramsey function and the Erd\H{o}s--Rogers function. Establishing a…

The domatic number of a graph $G$, denoted $dom(G)$, is the maximum possible cardinality of a family of disjoint sets of vertices of $G$, each set being a dominating set of $G$. It is well known that every graph without isolated vertices…

The induced arboricity of a graph $G$ is the smallest number of induced forests covering the edges of $G$. This is a well-defined parameter bounded from above by the number of edges of $G$ when each forest in a cover consists of exactly one…

We establish mild conditions under which a possibly irregular, sparse graph $G$ has "many" strong orientations. Given a graph $G$ on $n$ vertices, orient each edge in either direction with probability $1/2$ independently. We show that if…

An important theme of recent research in Ramsey theory has been establishing pseudorandomness properties of Ramsey graphs. An $N$-vertex graph is called $C$-Ramsey if it has no homogeneous set of size $C\log N$. A theorem of Bukh and…

This paper addresses the following question for a given graph $H$: what is the minimum number $f(H)$ such that every graph with average degree at least $f(H)$ contains $H$ as a minor? Due to connections with Hadwiger's Conjecture, this…

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every $2$-coloring of the…

The Ramsey number $r(G,H)$ is the minimum $N$ such that every graph on $N$ vertices contains $G$ as a subgraph or its complement contains $H$ as a subgraph. For integers $n \geq k \geq 1$, the $k$-book $B_{k,n}$ is the graph on $n$ vertices…

The Ramsey number $r(G)$ of a graph $G$ is the minimum $N$ such that every red-blue coloring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of $G$. Determining or estimating these numbers is one of the…

Let $f_\ell(n, k)$ denote the clique number of the xor-product of $\ell$ isomorphic Kneser graphs KG(n,k). Alon and Lubetzky investigated the case of complete graphs as a coding theory problem and showed $f_\ell(n,1)\leq \ell n +1$. Imolay,…

Given $r\in \mathbb{N}$ with $r\geq 4$, we show that there exists $n_0\in \mathbb{N}$ such that for every $n\geq n_0$, every $n$-vertex graph $G$ with $\delta(G)\geq (\frac{1}{2}+o(1))n$ and $\alpha_{r-2}(G)=o(n)$ contains a $K_{r}$-factor.…

Given a connected graph $G=(V,E)$ and a length function $\ell:E\to {\mathbb R}$ we let $d_{v,w}$ denote the shortest distance between vertex $v$ and vertex $w$. A $t$-spanner is a subset $E'\subseteq E$ such that if $d'_{v,w}$ denotes…

For $k \ge 1$ an integer, a set $S$ of vertices in a graph $G$ with minimum degree at least~$k-1$ is a $k$-tuple dominating set of $G$ if every vertex of $S$ is adjacent to at least $k-1$ vertices in $S$ and every vertex of $V(G) \setminus…

We investigate the expected value of various graph parameters associated with the minimum rank of a graph, including minimum rank/maximum nullity and related Colin de Verdi\`ere-type parameters. Let $G(v,p)$ denote the usual…