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The Lusternik-Schnirelmann theorem for graphs

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arxiv 1211.0750 v2 pith:NKRP2YDZ submitted 2012-11-05 math.AT cs.DMmath.GN

Frank Josellis , Oliver Knill This is my paper · ORCID
classification math.ATcs.DMmath.GN
keywords graphsminimaltcatcritquantityequalgraphhomotopic
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We prove the discrete Lusternik-Schnirelmann theorem telling that tcat(G) less or equal to crit(G) for a general simple graph G=(V,E). It relates the minimal number tcat(G) of in G contractible graphs covering G, with crit(G), the minimal number of critical points which an injective function f on the vertex set V can have. We also prove that the cup length cup(G) is less or equal to tcat(G) which is valid also for any finite simple graph. If cat(G) is the minimal tcat(H) among all graphs H homotopic to G and cri(G) is the minimal crit(H) among all graphs H homotopic to G, we get a relation between three homotopy invariants: an algebraic quantity (cup), a topological quantity (cat) and an analytic quantity (cri).

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  1. Elements of finite geometry I

    math.HO 2026-08 unverdicted novelty 2.0 of 10

    A review-style snapshot of twelve finite-geometry theorems, each claiming a discrete analogue of a well-known continuum result.

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