On the flexibility of toroidal embeddings

    Authors

    N. Robertson; X. Y. Zha;Y. Zhao

    Comments

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    Abbreviated Journal Title

    J. Comb. Theory Ser. B

    Keywords

    embedding; torus; flexibility; representativity; GRAPHS; UNIQUENESS; WIDTH; Mathematics

    Abstract

    Two embeddings psi(1) and psi(2) of a graph G in a surface Sigma are equivalent if there is a homeomorphism of Sigma to itself carrying psi(1) to psi(2). In this paper, we classify the flexibility of ernbeddings in the torus with representativity at least 4. We show that if a 3-connected graph G has an embedding psi in the torus with representativity at least 4, then one of the following holds: (i) psi is the unique embedding of G in the torus; (ii) G has three nonequivalent embeddings in the torus, G is the 4-cube Q4 (or C4 x C4), and each embedding of G forms a 4-by-4 toroidal grid; (iii) G has two nonequivalent embeddings in the torus, and G can be obtained from a toroidal 4-by-4 grid (faces are 2-colored) by splitting i (i < = 16) vertices along one-colored faces and replacing j (j < = 16) other colored faces with planar patches. (c) 2007 Elsevier Inc. All rights reserved.

    Journal Title

    Journal of Combinatorial Theory Series B

    Volume

    98

    Issue/Number

    1

    Publication Date

    1-1-2008

    Document Type

    Article

    Language

    English

    First Page

    43

    Last Page

    61

    WOS Identifier

    WOS:000252022400004

    ISSN

    0095-8956

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