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A conjecture on biconnected graphs and regular 3-cell complexes. (English) Zbl 1064.05047
The paper investigates the relationships between the notion of spatial graph (introduced by F. Luccio and L. Pagli [Congr. Numerantium 110, 33–41 (1995; Zbl 0905.05027)]) and the classical notion of CW-complex [see, for example, A. T. Lundell and S. Weingram, The topology of CW complexes (The University Series in Higher Mathematics, Van Nostrand Reinhold Company, New York etc.) (1969; Zbl 0207.21704)]. In particular, the spatiality degree of a graph \(G=(V,E)\) is shown to be the maximum number of 3-cells that a CW-complex with a 1-dimensional skeleton representing \(G\) can have on the three-dimensional compactified Euclidean space with its standard topology, assuming that two distinct 2-cells cannot share the same boundary and the two-dimensional skeleton is regular. Euler characteristic computation easily proves \(2(| E| -| V| )\) to be the upper bound for the spatiality degree of a biconnected graph. Based on some partial results, the author formulates the conjecture that for any biconnected graph \(G=(V,E)\) with at least four simple cycles there exists a regular three-dimensional CW-complex with a 1-dimensional skeleton representing \(G\) which has \(2(| E| -| V| )\) 3-cells and each cell has a distinct boundary.

05C10 Planar graphs; geometric and topological aspects of graph theory
57M15 Relations of low-dimensional topology with graph theory
57Q05 General topology of complexes