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- W1606228479 abstract "The neighborhood complexes of graphs were introduced by Lovasz in his proof of the Kneser conjecture. He showed that a certain topological property of $N(G)$ gives a lower bound for the chromatic number of $G$. In this paper, we study a combinatorial description of the fundamental groups of the neighborhood complexes. For a positive integer $r$, we introduce the $r$-fundamental group $pi_1^r(G,v)$ of a based graph $(G,v)$ and the $r$-neighborhood complex $N_r(G)$ of $G$. The $1$-neighborhood complex is the neighborhood complex. We show that the even part $pi_1^{2r}(G,v)_{ev}$, which is a subgroup of $pi_1^{2r}(G,v)$ with index 1 or 2, is isomorphic to the fundamental group of $(N_r(G),v)$ if $v$ is not isolated. We can use the $r$-fundamental groups to show the non-existence of graph homomorphisms. For example, we show that $pi_1^3(KG_{2k+1,k})$ is isomorphic to $mathbb{Z} /2$, and this implies that there is no graph homomorphism from $KG_{2k+1,k}$ to the 5-cycle graph $C_5$. We discuss the covering maps associated to $r$-fundamental groups." @default.
- W1606228479 created "2016-06-24" @default.
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- W1606228479 date "2012-10-10" @default.
- W1606228479 modified "2023-09-27" @default.
- W1606228479 title "Fundamental groups of neighborhood complexes" @default.
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