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On the parity of crossing numbers
Author(s) -
Archdeacon Dan,
Richter R. Bruce
Publication year - 1988
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.3190120302
Subject(s) - crossing number (knot theory) , combinatorics , mathematics , parity (physics) , graph , constant (computer programming) , integer (computer science) , discrete mathematics , physics , computer science , particle physics , engineering , programming language , intersection (aeronautics) , aerospace engineering
For an integer n ⩾ 1, a graph G has an n‐ constant crossing number if, for any two good drawings ϕ and ϕ′ of G in the plane, μ(ϕ) ≡ μ(ϕ′) (mod n ), where μ(ϕ) is the number of crossings in ϕ. We prove that, except for trivial cases, a graph G has n ‐constant crossing number if and only if n = 2 and G is either K p or K q,r , where p, q , and r are odd.
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