Abstract
In this paper we introduce a theory of edge shelling of graphs. Whereas the standard notion of shelling a simplicial complex involves a sequential removal of maximal simplexes, edge shelling involves a sequential removal of the edges of a graph. A necessary and sufficient condition for edge shellability is given in the case of 3-colored graphs, and it is conjectured that the result holds in general. Questions about shelling, and the dual notion of closure, are motivated by topological problems. The connection between graph theory and topology is by way of a complex ΔG associated with a graph G. In particular, every closed 2- or 3-manifold can be realized in this way. If ΔG is shellable, then G is edge shellable, but not conversely. Nevertheless, the condition that G is edge shellable is strong enough to imply that a manifold ΔG must be a sphere. This leads to completely graph-theoretic generalizations of the classical Poincaré Conjecture.
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Vince, A. Graphic matroids, shellability and the Poincare Conjecture. Geom Dedicata 14, 303–314 (1983). https://doi.org/10.1007/BF00146910
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DOI: https://doi.org/10.1007/BF00146910