### 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) |
---|---|

Pages (from-to) | 269-289 |

Number of pages | 21 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 85 |

Issue number | 2 |

DOIs | |

State | Published - 2002 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

**Families of regular graphs in regular maps.** / Wilson, Stephen E.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 85, no. 2, pp. 269-289. https://doi.org/10.1006/jctb.2001.2103

}

TY - JOUR

T1 - Families of regular graphs in regular maps

AU - Wilson, Stephen E

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0036318373&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036318373&partnerID=8YFLogxK

U2 - 10.1006/jctb.2001.2103

DO - 10.1006/jctb.2001.2103

M3 - Article

VL - 85

SP - 269

EP - 289

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 2

ER -