Homotopy type of neighborhood complexes of Kneser graphs, $K G_{2,k}$

Proceedings of the Indian Academy of Sciences. Mathematical sciences

NANDINI NILAKANTAN^1¹ 

[1] Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208 016, India^1

关键词: Hom complexes;    Kneser graphs;    discrete Morse theory;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Schrijver (Nieuw Archief voor Wiskunde, 26(3) (1978) 454–461) identified a family of vertex critical subgraphs of the Kneser graphs called the stable Kneser graphs $SG_{n,k}$ . Björner and de Longueville (Combinatorica 23(1) (2003) 23–34) proved that the neighborhood complex of the stable Kneser graph $SG_{n,k}$ is homotopy equivalent to a$k$-sphere. In this article, we prove that the homotopy type of the neighborhood complex of the Kneser graph $K G_{2,k}$ is a wedge of $(k + 4)(k + 1) + 1$ spheres of dimension $k$. We construct a maximal subgraph $S_{2,k}$ of $K G_{2,k}$ , whose neighborhood complex is homotopy equivalent to the neighborhood complex of $SG_{2,k}$ . Further, we prove that the neighborhood complex of $S_{2,k}$ deformation retracts onto the neighborhood complex of $SG_{2,k}$ .

【 授权许可】

CC BY