Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-23T18:48:12.137Z Has data issue: false hasContentIssue false
  • English
  • Français

Smallest regular graphs without near 1-factors

Part of: Graph theory

Published online by Cambridge University Press:  09 April 2009

Gary Chartrand ,
Sergio Ruiz  and
Curtiss E. Wall
Gary Chartrand
Affiliation:
Department of Mathematics, Western Michigan University, Kalamazoo, Michigan 49008, U.S.A.
Sergio Ruiz
Affiliation:
Office of Courseware Development, Norfolk Public Schools, Norfolk, Virginia 23508, U.S.A.
Curtiss E. Wall
Affiliation:
Instituto de Matemáticas, Universided Católica de Valparaíso, Valparaíso, Chile
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A near 1-factor of a graph of order 2n ≧ 4 is a subgraph isomorphic to (n − 2) K2 ∪ P3 ∪ K1. Wallis determined, for each r ≥ 3, the order of a smallest r-regular graph of even order without a 1-factor; while for each r ≧ 3, Chartrand, Goldsmith and Schuster determined the order of a smallest r-regular, (r − 2)-edge-connected graph of even order without a 1-factor. These results are extended to graphs without near 1-factors. It is known that every connected, cubic graph with less than six bridges has a near 1-factor. The order of a smallest connected, cubic graph with exactly six bridges and no near 1-factor is determined.

MSC classification

Secondary: 05C99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 2 , October 1986 , pp. 193 - 210
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Chartrand, G., Goldsmith, D. L. and Schuster, S., ‘A sufficient condition for graphs with 1-factors’, Colloq. Math. 41 (1979), 173–178.CrossRefGoogle Scholar
[2]Chartrand, G., Kapoor, S. F., Lesniak, L. and Schuster, S., ‘Near 1-factors in graphs’ (Proceedings of the fourteenth southeastern conference on combinatorics, graph theory and computing, to appear).Google Scholar
[3]Petersen, J., ‘Die Theorie der regulären Graphen’, Acta Math. 15 (1981), 163–220.Google Scholar
[4]Pila, J., ‘Connected regular graphs without one factors’, Ars Combinatoria 18 (1984), 161–172.Google Scholar
[5]Tutte, W. T., ‘The factorizations of linear graphs’, J. London Math. Soc. 22 (1947), 107–111.CrossRefGoogle Scholar
[6]Wallis, W. D., ‘The smallest regular graphs without one-factors’, Ars Combinatoria 11 (1981), 295–300.Google Scholar