Related papers: Independence Complexes of Stable Kneser Graphs

Schrijver identified a family of vertex critical subgraphs of the Kneser graphs called the stable Kneser graphs $SG_{n,k}$. Bj\"{o}rner and de Longueville proved that the neighborhood complex of the stable Kneser graph $SG_{n,k}$ is…

The stable Kneser graph $SG_{n,k}$, $n\ge1$, $k\ge0$, introduced by Schrijver \cite{schrijver}, is a vertex critical graph with chromatic number $k+2$, its vertices are certain subsets of a set of cardinality $m=2n+k$. Bj\"orner and de…

We denote by SG_{n,k} the stable Kneser graph (Schrijver graph) of stable n-subsets of a set of cardinality 2n+k. For k congruent 3 (mod 4) and n\ge2 we show that there is a component of the \chi-colouring graph of SG_{n,k} which is…

In 2002, A. Bj\"orner and M. de Longueville showed the neighborhood complex of the $2$-stable Kneser graph ${KG(n, k)}_{2-\textit{stab}}$ has the same homotopy type as the $(n-2k)$-sphere. A short time ago, an analogous result about the…

In 2003, A. Bjorner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SG_{n,k} is homotopy equivalent to a k-sphere. Further, for n=2 they showed that the neighborhood complex deformation retracts to a…

For integers $k\geq 1$ and $n\geq 2k+1$, the Schrijver graph $S(n,k)$ has as vertices all $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ that contain no two cyclically adjacent elements, and an edge between any two disjoint sets. More…

For $k\geq 1$ and $n\geq 2k$, the Kneser graph $KG(n,k)$ has all $k$-element subsets of an $n$-element set as vertices; two such subsets are adjacent if they are disjoint. It was first proved by Lov\'{a}sz that the chromatic number of…

A Kneser graph $KG_{n,k}$ is a graph whose vertices are in one-to-one correspondence with $k$-element subsets of $[n],$ with two vertices connected if and only if the corresponding sets do not intersect. A famous result due to Lov\'asz…

For positive integers k,n, we investigate the simplicial complex NM_k(n) of all graphs G on vertex set [n] such that every matching in G has size less than k. This complex (along with other associated cell complexes) is found to be homotopy…

The independence complex of a graph is a simplicial complex whose faces correspond to the independent sets of $G$. While independence complexes have been studied extensively for many graph classes, including square grid graphs, relatively…

In this article, we prove that the neighborhood complex of the Kneser graph $KG_{3,k}$ is of the same homotopy type as that of a wedge of $\frac{(k+1)(k+3)(k+4)(k+6)}{4}+1$ spheres of dimension $k$. We construct a maximal subgraph $S_{3,k}$…

For $r\geq 1$, the $r$-independence complex of a graph $G$ is a simplicial complex whose faces are subset $I \subseteq V(G)$ such that each component of the induced subgraph $G[I]$ has at most $r$ vertices. In this article, we determine the…

The Kneser graph $K(n,k)$ is defined for integers $n$ and $k$ with $n \geq 2k$ as the graph whose vertices are all the $k$-subsets of $[n]=\{1,2,\ldots,n\}$ where two such sets are adjacent if they are disjoint. The Schrijver graph $S(n,k)$…

The Kneser graph $KG_{n,k}$ is the graph whose vertices are the $k$-element subsets of $[n],$ with two vertices adjacent if and only if the corresponding sets are disjoint. A famous result due to Lov\'asz states that the chromatic number of…

For $k,s\geq2$, the $s$-stable Kneser graphs are the graphs with vertex set the $k$-subsets $S$ of $\{1,\ldots,n\}$ such that the circular distance between any two elements in $S$ is at least $s$ and two vertices are adjacent if and only if…

Consider classical Kneser's graph $K(n,r)$: for two natural numbers $ r, n $ such that $r \le n / 2$, its vertices are all the subsets of $[n]=\{1,2,\ldots,n\}$ of size $r$, and two such vertices are adjacent if the corresponding subsets…

The independence complex of a graph G is a simplicial complex whose simplices are the independent sets in G. In the last couple of decades, the independence complexes of square grids (with various boundary conditions) have gained much…

This paper is devoted to the neighborhood complexes of the induced $k$-independent graphs. Inspired by the surprising correspondence between total $k$-cut complex of $n$-cycle $C_n$ and neighborhood complex of stable Kneser graph $SG(n,k)$,…

The Kneser signed graph $\KS(n,k)$, $k\leq n$, is the graph whose vertices are signed $k$-subsets of $[n]$ (i.e. $k$-subsets $S$ of $\{ \pm 1, \pm 2, \ldots, \pm n\}$ such that $S\cap (-S)=\emptyset$). Two vertices $A$ and $B$ are adjacent…

The Kneser Graph $K(n,k)$ has as vertices all $k$-subsets of $\{1,\ldots,n\}$ and edges connecting two vertices if they are disjoint. The $s$-stable Kneser Graph $K_{s-stab}(n, k)$ is obtained from the Kneser graph by deleting vertices with…