8 research outputs found

    Generalization of neighborhood complexes

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    We introduce the notion of r-neighborhood complex for a positive integer r, which is a natural generalization of Lovasz neighborhood complex. The topologies of these complexes give some obstructions of the existence of graph maps. We applied these complexes to prove the nonexistence of graph maps about Kneser graphs. We prove that the fundamental groups of r-neighborhood complexes are closely related to the (2r)-fundamental groups defined in the author's previous paper.Comment: 8 page

    Equivariant Piecewise-Linear Topology and Combinatorial Applications

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    For G a finite group, we develop some theory of G-equivariant piecewise-linear topology and prove characterization theorems for G-equivariant regular neighborhoods. We use these results to prove a conjecture of Csorba that the Lovász complex Hom(C5,Kn) of graph multimorphisms from the 5-cycle C5 to the complete graph Kn is equivariantly homeomorphic to the Stiefel manifold, Vn-1,2, the space of (ordered) orthonormal 2-frames in Rn-1 with respect to an action of the cyclic group of order 2

    The equivariant topology of stable Kneser graphs

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    The stable Kneser graph SGn,kSG_{n,k}, nβ‰₯1n\ge1, kβ‰₯0k\ge0, introduced by Schrijver \cite{schrijver}, is a vertex critical graph with chromatic number k+2k+2, its vertices are certain subsets of a set of cardinality m=2n+km=2n+k. Bj\"orner and de Longueville \cite{anders-mark} have shown that its box complex is homotopy equivalent to a sphere, \Hom(K_2,SG_{n,k})\homot\Sphere^k. The dihedral group D2mD_{2m} acts canonically on SGn,kSG_{n,k}, the group C2C_2 with 2 elements acts on K2K_2. We almost determine the (C2Γ—D2m)(C_2\times D_{2m})-homotopy type of \Hom(K_2,SG_{n,k}) and use this to prove the following results. The graphs SG2s,4SG_{2s,4} are homotopy test graphs, i.e. for every graph HH and rβ‰₯0r\ge0 such that \Hom(SG_{2s,4},H) is (rβˆ’1)(r-1)-connected, the chromatic number Ο‡(H)\chi(H) is at least r+6r+6. If kβˆ‰{ 0,1,2,4,8 }k\notin\set{0,1,2,4,8} and nβ‰₯N(k)n\ge N(k) then SGn,kSG_{n,k} is not a homotopy test graph, i.e.\ there are a graph GG and an rβ‰₯1r\ge1 such that \Hom(SG_{n,k}, G) is (rβˆ’1)(r-1)-connected and Ο‡(G)<r+k+2\chi(G)<r+k+2.Comment: 34 pp

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum
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