Vol. 76, No. 2, 1978

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The non-orientable genus of the n-cube

Mark Jungerman

Vol. 76 (1978), No. 2, 443–451
Abstract

For the purposes of embedding theory, a graph consists of a collection of points, called vertices, certain pairs of which are joined by homeomorphs of the unit interval, called edges. Edges may intersect only at vertices, and no vertex is contained in the interior of an edge. The graph thus becomes a topological space as a subspace of R3. An embedding of a graph G in a compact 2-manifold (surface) S is then just an embedding of G in S as a topological space. The genus, γ(G), of G is the minimum genus among all orientable surfaces into which G may be embedded. The non-orientable genus, γ(G), is defined analogously. The n-cube, Qn, is a well known graph wnich generalizes the square and the standard cube. In this paper the following formula is proven:

Theorem.

        (
|{2 + 2n−2(n − 4) n ≧ 6
γ(Qn) =  3 + 2n−2(n − 4) n = 4,5
|(
1              n ≦ 3.

Mathematical Subject Classification 2000
Primary: 05C10
Secondary: 57M15
Milestones
Received: 20 April 1977
Published: 1 June 1978
Authors
Mark Jungerman