Neighborhood complexes of some exponential graphs. (English) Zbl 1336.05045
Summary: 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 \(\mathrm{Hom}(K_2 \times K_{n}, K_{m})\) is homotopic to \(S^{m-2}\), if \(2\leqslant m <n\).
MSC:
| 05C15 | Coloring of graphs and hypergraphs |
| 57M15 | Relations of low-dimensional topology with graph theory |
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