O ct 2 00 5 Combinatorial groupoids , cubical complexes , and the Lovász conjecture
نویسنده
چکیده
A foundation is laid for a theory of combinatorial groupoids, allowing us to use concepts like “holonomy”, “parallel transport”, “bundles”, “combinatorial curvature” etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes and other combinatorial objects. A new, holonomy-type invariant for cubical complexes is introduced, leading to a combinatorial “Theorema Egregium” for cubical complexes nonembeddable into cubical lattices. Parallel transport of Hom-complexes and maps is used as a tool for extending Babson-Kozlov-Lovász graph coloring results to more general statements about non-degenerate maps (colorings) of simplicial complexes and graphs.
منابع مشابه
Combinatorial Groupoids, Cubical Complexes, and the Lovász Conjecture
A foundation is laid for a theory of combinatorial groupoids, allowing us to use concepts like “holonomy”, “parallel transport”, “bundles”, “combinatorial curvature” etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes and other combinatorial objects. A new, holonomy-type invariant for cubical complexes is introduced, leading to a combinatorial “Theorema Egregium”...
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متن کاملar X iv : m at h / 04 10 33 5 v 1 [ m at h . C O ] 1 4 O ct 2 00 4 HIGHER CONNECTIVITY OF GRAPH COLORING COMPLEXES SONJA
The main result of this paper is a proof of the following conjecture of Babson & Kozlov: Theorem. Let G be a graph of maximal valency d, then the complex Hom (G, Kn) is k-connected, whenever n ≥ d + k + 2. Here Hom (−,−) denotes the polyhedral complex introduced by Lovász to study the topological lower bounds for chromatic numbers of graphs. We will also prove, as a corollary to the main theore...
متن کاملar X iv : m at h / 04 02 39 5 v 3 [ m at h . C O ] 1 8 Ju l 2 00 5 PROOF OF THE LOVÁSZ CONJECTURE
To any two graphs G and H one can associate a cell complex Hom (G,H) by taking all graph multihomorphisms from G to H as cells. In this paper we prove the Lovász Conjecture which states that if Hom (C2r+1, G) is k-connected, then χ(G) ≥ k + 4, where r, k ∈ Z, r ≥ 1, k ≥ −1, and C2r+1 denotes the cycle with 2r + 1 vertices. The proof requires analysis of the complexes Hom (C2r+1,Kn). For even n,...
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تاریخ انتشار 2005