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Drawing orders with few slopes. (English) Zbl 0731.06007

Authors’ summary: “It is common to draw a diagram of an ordered set with as few slopes as seem possible; the maximum number of upper covers or lower covers of an element is an obvious lower bound to the number of different slopes needed. We construct lattices with at most two (respectively, three) upper and lower covers which require at least three (respectively, four) different slopes - despite a conjecture of B. Sands to the contrary. Moreover, we characterize lattices with two-slope diagrams. It follows, for example, that every planar lattice with at most two upper and two lower covers has a (planar) two-slope diagram.”

MSC:

06A07 Combinatorics of partially ordered sets
05C10 Planar graphs; geometric and topological aspects of graph theory
06B99 Lattices
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References:

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