Hostname: page-component-5c6d5d7d68-txr5j Total loading time: 0 Render date: 2024-08-15T13:00:10.181Z Has data issue: false hasContentIssue false
  • English
  • Français

k-Degenerate Graphs

Published online by Cambridge University Press:  20 November 2018

Don R. Lick  and
Arthur T. White
Rights & Permissions [Opens in a new window]

Extract

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.

Graphs possessing a certain property are often characterized in terms of a type of configuration or subgraph which they cannot possess. For example, a graph is totally disconnected (or, has chromatic number one) if and only if it contains no lines; a graph is a forest (or, has point-arboricity one) if and only if it contains no cycles. Chartrand, Geller, and Hedetniemi [2] defined a graph to have property Pn if it contains no subgraph homeomorphic from the complete graph Kn+1 or the complete bipartite graph

For the first four natural numbers n, the graphs with property Pn are exactly the totally disconnected graphs, forests, outerplanar and planar graphs, respectively. This unification suggested the extension of many results known to hold for one of the above four classes of graphs to one or more of the remaining classes.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 5 , 01 October 1970 , pp. 1082 - 1096
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Behzad, M. and Chartrand, G., An introduction to the theory of graphs (Allyn and Bacon, to appear).Google Scholar
2. Chartrand, G., Geller, D., and Hedetniemi, S., Graphs with forbidden subgraphs, J. Combinatorial Theory (to appear).Google Scholar
3. Chartrand, G. and Kronk, H., The point-arboricity of planar graphs, J. London Math. Soc. 44 (1969), 612–616.Google Scholar
4. Chartrand, G., Kronk, H., and Wall, C., The point-arboricity of a graph, Israel J. Math. 6 (1968), 169–175.Google Scholar
5. Dirac, G. A., A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. Soc. 27 (1952), 85–92.Google Scholar
6. Dirac, G. A., Some theorems on abstract graphs, Proc. London Math. Soc, Ser. 3, 2 (1952), 69–81.Google Scholar
7. Dirac, G. A., The structure of k-chromatic graphs, Fund. Math. 40 (1953), 42–55.Google Scholar
8. Harary, F., Graph theory (Addison-Wesley, Reading, Massachusetts, 1969).Google Scholar
9. Lick, D. R. and White, A. T., Chromatic-durable graphs (submitted for publication).Google Scholar
10. Mitchem, J., On extremal partitions of graphs, Thesis, Western Michigan University, Kalamazoo, Michigan, 1970.Google Scholar
11. Szekeres, G. and Wilf, H. S., An inequality for the chromatic number of a graph, J. Combinatorial Theory 4 (1968), 1–3.Google Scholar
12. Wilf, H. S., The eigenvalues of a graph and its chromatic number, J. London Math. Soc. 42 (1967), 330–332.Google Scholar