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Nonexistence of a weakly neighbourly polyhedral map of type $$\{6,6\}$$. (English) Zbl 1080.52506
For an $$n$$-vertex polyhedral map of type $$\{p,p\}$$ the inequality $$n\geq(p-1)^2$$ holds with equality precisely for the class of weakly neighbourly maps, that is, maps with the minimum number of edges. For $$p\leq 5$$ it is known that there are such weakly neighbourly maps.
In the present paper it is shown by a computer algorithm that there is none for $$p=6$$. Therefore, Brehm’s example of a map of type $$\{6,6\}$$ with $$n=26$$ is best possible, see U. Brehm in [Topics in combinatorics and graph theory, 153–162 (1990; Zbl 0703.57008)].
##### MSC:
 52B70 Polyhedral manifolds 51M20 Polyhedra and polytopes; regular figures, division of spaces 57M20 Two-dimensional complexes (manifolds) (MSC2010) 57M15 Relations of low-dimensional topology with graph theory 52-04 Software, source code, etc. for problems pertaining to convex and discrete geometry
edge graph
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##### References:
 [1] Brehm U., Topics in Comb, and Graph Theory (Ringel-Fetstschrift) pp 153– (1990) [2] Brehm U., Beiträge zur Algebra und Geometrie 43 pp 583– (2002) [3] Brehm U., Handbooks of Discrete and Computational Geometry pp 345– (1997) [4] Brehm U., Handbook of Convex Geometry pp 535– (1993) [5] Datta B., Discrete and Comput Geom. 26 pp 429– (2001)
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