Abstract
The purpose of this paper is to investigate some properties of the crossing number χ(P) of a posetP. We first study the crossing numbers of the product and the lexicographical sum of posets. The results are similar to the dimensions of these posets. Then we consider the problem of what happens to the crossing number when a point is taken away from a poset. We show that ifP is a poset such that χ∈P and χ(P−χ)⩾1, then 1/2 χ(P)⩽χ(P−χ)⩽χ(P). We don't know yet how to improve the lower bound. We also determine the crossing numbers of some subposets of the Boolean latticeB n which consist of some specified ranks. Finally we show that Ψ n is crossing critical where Ψ n is the subposet ofB n which is restricted to rank 1, rankn−1 and middle rank(s). Some open problems are raised at the end of this paper.
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Communicated by I. Rival
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Lin, C. The crossing number of posets. Order 11, 169–193 (1994). https://doi.org/10.1007/BF01108601
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DOI: https://doi.org/10.1007/BF01108601