On the parity of crossing numbers
Dan Archdeacon, R. Bruce Richter · Journal of Graph Theory · 1988
Abstract 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.