### Abstract

The question of when a given graph can be the underlying graph of a regular map has roots a hundred years old and is currently the object of several threads of research. This paper outlines this topic briefly and proves that a product of graphs which have regular embeddings also has such an embedding. We then present constructions for members of three families: (1) circulant graphs, (2) wreath graphs W(k, n), whose vertices are ordered pairs (i, j), 0 ≤ i < k, 0 ≤ j < n, and whose edges are all possible (i, j) - (i + 1, j′), and (3) depleted wreath DW(k, n), the subgraph of W(k, n) left by removing all edges in which j = j′. We open the question of multiplicity of occurrence and we list the underlying graphs of rotary maps with no more than 50 edges.

Original language | English (US) |
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Pages (from-to) | 269-289 |

Number of pages | 21 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 85 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2002 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics