The LusternikSchnirelmann theorem for graphs
Abstract
We prove the discrete LusternikSchnirelmann 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).
 Publication:

arXiv eprints
 Pub Date:
 November 2012
 DOI:
 10.48550/arXiv.1211.0750
 arXiv:
 arXiv:1211.0750
 Bibcode:
 2012arXiv1211.0750J
 Keywords:

 Mathematics  Algebraic Topology;
 Computer Science  Discrete Mathematics;
 Mathematics  General Topology;
 55M30;
 58E05;
 05C75;
 05C10;
 57M15;
 57Q10
 EPrint:
 29 pages, 7 figures. Main results unchanged but cat(G) had not yet been homotopy invariant. 3 more references, smaller typos and a figure correction