Families of regular graphs in regular maps

Research output: Contribution to journalArticle

6 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)269-289
Number of pages21
JournalJournal of Combinatorial Theory. Series B
Volume85
Issue number2
DOIs
StatePublished - 2002

Fingerprint

Regular Map
Regular Graph
Graph in graph theory
Regular Embedding
Circulant Graph
Ordered pair
Thread
Subgraph
Multiplicity
Family
Roots

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

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

In: Journal of Combinatorial Theory. Series B, Vol. 85, No. 2, 2002, p. 269-289.

Research output: Contribution to journalArticle

@article{1378885a5bb440bf9f67ecea95bc6a1d,
title = "Families of regular graphs in regular maps",
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.",
author = "Wilson, {Stephen E}",
year = "2002",
doi = "10.1006/jctb.2001.2103",
language = "English (US)",
volume = "85",
pages = "269--289",
journal = "Journal of Combinatorial Theory. Series B",
issn = "0095-8956",
publisher = "Academic Press Inc.",
number = "2",

}

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

AN - SCOPUS:0036318373

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 -