Related papers: Independent sets in subgraphs of a shift graph

For a graph $G$, a vertex subset $S$ is called a maximum generalized $k$-independent set if the induced subgraph $G[S]$ does not contain a $k$-tree as its subgraph, and the subset has maximum cardinality. The generalized $k$-independence…

The size of a largest independent set of vertices in a given graph $G$ is denoted by $\alpha(G)$ and is called its independence number (or stability number). Given a graph $G$ and an integer $K,$ it is NP-complete to decide whether…

Let $G = (V,E)$ be a graph and $k \ge 0$ an integer. A $k$-independent set $S \subseteq V$ is a set of vertices such that the maximum degree in the graph induced by $S$ is at most $k$. With $\alpha_k(G)$ we denote the maximum cardinality of…

Let $G$ be a graph on $n$ vertices of independence number $\alpha(G)$ such that every induced subgraph of $G$ on $n-k$ vertices has an independent set of size at least $\alpha(G) - \ell$. What is the largest possible $\alpha(G)$ in terms of…

We describe an infinite family of graphs $G_n$, where $G_n$ has $n$ vertices, independence number at least $n/4$, and no set of less than $\sqrt{n}/2$ vertices intersects all its maximum independent sets. This is motivated by a question of…

The {\em independence ratio} of a graph $G$ is defined by \[ \iota(G) := \sup_{X \subset V(G)} \frac{|X|}{\alpha(X)},\] where $\alpha(X)$ is the independence number of the subgraph of $G$ induced by $X$. The independence ratio is a…

Let $G$ be a graph on $n$ vertices, independence number $\alpha(G)$, Lov\'asz theta function $\vartheta(G)$, and Shannon capacity $\Theta(G)$. We define $n_{\ge0}(G)$ to be the minimum number of non-negative eigenvalues taken over all…

The following natural problem was raised independently by Erd\H{o}s-Hajnal and Linial-Rabinovich in the late 80's. How large must the independence number $\alpha(G)$ of a graph $G$ be whose every $m$ vertices contain an independent set of…

Erd\H{o}s and Hajnal conjectured that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or a stable set of size at least $|G|^c$ (a graph is $H$-free if it has no induced subgraph isomorphic to $H$).…

Given $d>0$ and a positive integer $n$, let $G$ be a triangle-free graph on $n$ vertices with average degree $d$. With an elegant induction, Shearer (1983) tightened a seminal result of Ajtai, Koml\'os and Szemer\'edi (1980/1981) by proving…

A seminal result of Hajnal and Szemer\'{e}di states that if a graph $G$ with $n$ vertices has minimum degree $\delta(G) \ge (r-1)n/r$ for some integer $r \ge 2$, then $G$ contains a $K_r$-factor, assuming $r$ divides $n$. Extremal examples…

The ultimate independence ratio of a graph $G$ is defined as $\mathscr{I}(G) = \lim_{k\rightarrow\infty } \frac{\alpha(G^{\Box k})}{|V(G)|^k},$ where $\alpha(G^{\Box k})$ is the independence number of the Cartesian product of $k$ copies of…

Let $\Gamma$ be a function that maps two arbitrary graphs $G$ and $H$ to a non-negative real number such that $$\alpha(G^{\boxtimes n})\leq \alpha(H^{\boxtimes n})\Gamma(G,H)^n$$ where $n$ is any natural number and $G^{\boxtimes n}$ is the…

An $n$-vertex, $d$-regular graph can have at most $2^{n/2+o_d(n)}$ independent sets. In this paper we address what happens with this upper bound when we impose the further condition that the graph has independence number at most $\alpha$.…

Settling Kahn's conjecture (2001), we prove the following upper bound on the number $i(G)$ of independent sets in a graph $G$ without isolated vertices: \[ i(G) \le \prod_{uv \in E(G)} i(K_{d_u,d_v})^{1/(d_u d_v)}, \] where $d_u$ is the…

Let $G=(V(G),E(G))$ be a graph with set of vertices $V(G)$ and set of edges $E(G)$. For $k\ge 0$ an integer, a subset $I_k$ of $V(G)$ is called a $k$-nearly independent vertex subset of $G$ if $I_k$ induces a subgraph of size $k$ in $G$.…

Let $G$ be a $K_4$-free graph of order $n$ and let $k$ be an integer with $0\leq k\leq n$. We show the existence of positive constants $\eta$ and $\nu$ such that $G$ has at most $(4-\eta)^{(5-\eta)k-n}(5-\eta)^{n-(4-\eta)k}$ maximal…

Let $G$ be a graph and $\mathcal{F}$ a family of graphs. Define $\alpha_{\mathcal{F}}(G)$ as the maximum order of any induced subgraph of $G$ that belongs to the family $\mathcal{F}$. For the family $\mathcal{F}$ of graphs with…

Write ${\cal I}(G)$ for the set of independent sets of a graph $G$ and $i(G)$ for $|{\cal I}(G)|$. It has been conjectured (by Alon and Kahn) that for an $N$-vertex, $d$-regular graph $G$, $$ i(G) \leq \left(2^{d+1}-1\right)^{N/2d}. $$ If…

A graph $G$ is well-covered if all its maximal stable sets have the same size, denoted by alpha(G) (M. D. Plummer, 1970). If for any $k$ the $k$-th coefficient of a polynomial I(G;x) is equal to the number of stable sets of cardinality $k$…