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k-Degenerate Graphs

  • Don R. Lick and Arthur T. White
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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 P n if it contains no subgraph homeomorphic from the complete graph K n+1 or the complete bipartite graph

For the first four natural numbers n, the graphs with property P n 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.

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References
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1. Behzad, M. and Chartrand, G., An introduction to the theory of graphs (Allyn and Bacon, to appear).
2. Chartrand, G., Geller, D., and Hedetniemi, S., Graphs with forbidden subgraphs, J. Combinatorial Theory (to appear).
3. Chartrand, G. and Kronk, H., The point-arboricity of planar graphs, J. London Math. Soc. 44 (1969), 612616.
4. Chartrand, G., Kronk, H., and Wall, C., The point-arboricity of a graph, Israel J. Math. 6 (1968), 169175.
5. Dirac, G. A., A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. Soc. 27 (1952), 8592.
6. Dirac, G. A., Some theorems on abstract graphs, Proc. London Math. Soc, Ser. 3, 2 (1952), 6981.
7. Dirac, G. A., The structure of k-chromatic graphs, Fund. Math. 40 (1953), 4255.
8. Harary, F., Graph theory (Addison-Wesley, Reading, Massachusetts, 1969).
9. Lick, D. R. and White, A. T., Chromatic-durable graphs (submitted for publication).
10. Mitchem, J., On extremal partitions of graphs, Thesis, Western Michigan University, Kalamazoo, Michigan, 1970.
11. Szekeres, G. and Wilf, H. S., An inequality for the chromatic number of a graph, J. Combinatorial Theory 4 (1968), 13.
12. Wilf, H. S., The eigenvalues of a graph and its chromatic number, J. London Math. Soc. 42 (1967), 330332.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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