neighborhood complex
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2021 ◽  
Vol 344 (4) ◽  
pp. 112302
Author(s):  
Hamid Reza Daneshpajouh ◽  
József Osztényi

10.37236/7549 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Samir Shukla

To estimate the lower bound for the chromatic number of a graph $G$,  Lovász associated a simplicial complex  $\mathcal{N}(G)$ called the neighborhood complex and related the topological connectivity of $\mathcal{N}(G)$ to the chromatic number of $G$. More generally he proved that the chromatic number of $G$ is bounded below by the topological connectivity of $\mathcal{N}(G)$ plus $3$. In this article, we consider the graphs of maximal degree at most $3$ and $4$-regular circulant graphs. We show that each connected component of the neighborhood complexes of these graphs is homotopy equivalent either to a point, to a wedge sum of circles, to a wedge sum of $2$-spheres $S^2$, to $S^3$, to a garland of $2$-spheres $S^2$ or to a connected sum of tori. 


10.37236/5312 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Nandini Nilakantan ◽  
Samir Shukla

In this article, we consider the bipartite graphs $K_2 \times K_n$. We first show that the connectedness of the neighborhood complex $\mathcal{N}(K_{n+1}^{K_n}) =0$. Further, we show that Hom$(K_2 \times K_{n}, K_{m})$ is homotopic to $S^{m-2}$, if $2\leq m <n$.


2008 ◽  
Vol 51 (4) ◽  
pp. 535-544 ◽  
Author(s):  
Péter Csorba

AbstractWe prove that the neighborhood complex N(G), the box complex B(G), the homomorphism complex Hom(K2, G) and the Lovász complex L(G) have the same simple ℤ2-homotopy type in the sense of Whitehead. We show that these graph complexes are simple ℤ2-universal.


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