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The Slimming Number and Genus of Graphs

Published online by Cambridge University Press:  20 November 2018

Richard K. Guy
Richard K. Guy*
Affiliation:
University of Calgary, Calgary, Alberta
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J. Ch. Boland suggested, and Mrs. Sheehan named, the idea of the slimming number of a graph G, i.e. the minimum number, s(G), of edges, e 1e 2,…, e s, which must be removed from G in order that G—∪ e i be planar.

For the complete graph, K n (n≥3), it may be seen by Euler's formula that a planar subgraph contains at most 3n—6 edges; moreover one may construct such a subgraph inductively, starting from K 3, and adding points successively, joining them to the three vertices of the region in which they lie, so

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Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 2 , June 1972 , pp. 195 - 200
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Battle, J., Harary, F., and Kodama, Y., Every planar graph with nine points has a non-planar complement, Bull. Amer. Math. Soc. 68 (1962), 569-571; M.R. 27 (1964), #5248.Google Scholar
2. Beineke, L. W., Harary, F., and Moon, J. W., On the thickness of the complete bipartite graph. Proc. Cambridge Philos. Soc, 60 (1964), 1-5; M.R. 28 (1964), #1611.Google Scholar
3. Grünbaum, B., Convex Polytopes, Interscience, New York, (1967), 61,122.Google Scholar
4. Kuratowski, K., Sur le probl#ème des courbes gauches en topologie, Fund. Math., 15 (1930), 271-283.Google Scholar
5. Ringel, G., #Über drei kombinatorische Probl#ème am n-dimensionalen W#ürfel und W#ürfelgitter, Abh. Math. Sem. Univ. Hamburg, 20 (1955), 10-19; M.R. 17 (1956), #772.Google Scholar
6. Ringel, G., Das Geschlecht des vollst#ándigen paaren Graphen, Abh. Math. Sem. Univ. Hamburg, 28 (1965), 139-150; M.R. 32 (1966), #6439.Google Scholar
7. Ringel, G., F#árbungsprobleme auf Fl#áchen und Graphen, VE. Deutsche. Verlag. Berlin. 1959; M.R. 22 (1961), #A235.Google Scholar
8. Ringel, G., The genus of graphs, in R. K. Guy, et al. (editors), Combinatorial Structures and their Applications (Proc. Calgary Internat. Conference, 1969), Gordon and Breach, New York, (1970), 361-366.Google Scholar
9. Ringel, G. and Youngs, J. W. T., Losung des problems der Nachbargebiete auf orientierbaren Fl#áchen, Arch. Math. (Basel), 20 (1969), 190-201; M.R. 39 (1970), #4049.Google Scholar
10. Ringel, G. and Youngs, J. W. T., Remarks on the Heawood conjecture, in F. Harary, (editor), Proof Techniques in Graph Theory, Academic Press, New York, (1969), 133-138.Google Scholar
11. Ringel, G. and Youngs, J. W. T., Solution of the Heawood map-coloring problem—case 11, J. Combinatorial Theory, 7 (1969), 71-93; M.R. 39 (1970), #1360. —case 2, ibid., 342-352. ? case 8, ibid., 353-363.Google Scholar
12. Terry, C. M., Welch, L. R., and Youngs, J. W. T., The Genus ofKn, n = 12(2m), Bull. Amer. Math. Soc. 71 (1965), 653-656. Ditto, n = 12s, ibid., 657-660; M.R. 31 (1966), #4022a, b.Google Scholar
13. Terry, C. M., Welch, L. R., and Youngs, J. W. T., The genus of K12s, J. Combinatorial Theory, 2 (1967), 43-60; M.R. 34 (1967), #6755.Google Scholar
14. Terry, C. M., Welch, L. R., and Youngs, J. W. T., Solution of the Heawood map-coloring problem, case 4, J. Combinatorial Theory, 8 (1970), 170-174.Google Scholar
15. Tutte, W. T., On the non-biplanar character of the complete 9-graph, Canad. Math. Bull., 6 (1963), 319-330; M.R. 28 (1964), #2535.Google Scholar
16. Youngs, J. W. T., Minimal imbeddings and the genus of a graph, J. Math. Mech., 12 (1963), 303-315; M.R. 26 (1963), #3043.Google Scholar
17. Youngs, J. W. T., The Heawood map-coloring conjecture, in F. Harary (editor), Graph Theory and Theoretical Physics, Academic Press, London, 1967, 313-354; M.R. 38 (1969), #4357.Google Scholar
18. Youngs, J. W. T., Solution of the Heawood map-coloring problem, cases 3, 5, 6 and 9. J. Combinatorial Theory, 8 (1970), 175-219. cases 1, 7 and 10, ibid., 220-231.Google Scholar