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Longest Cycles in 3-Connected 3-Regular Graphs

  • J. A. Bondy (a1) and M. Simonovits (a2)
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In this paper, we study the following question: How long a cycle must there be in a 3-connected 3-regular graph on n vertices? For planar graphs this question goes back to Tait [6], who conjectured that any planar 3-connected 3-regular graph is hamiltonian. Tutte [7] disproved this conjecture by finding a counterexample on 46 vertices. Using Tutte's example, Grunbaum and Motzkin [3] constructed an infinite family of 3-connected 3-regular planar graphs such that the length of a longest cycle in each member of the family is at most nc, where c = 1 – 2–17 and n is the number of vertices. The exponent c was subsequently reduced by Walther [8, 9] and by Grùnbaum and Walther [4].

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References
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1. Barnette, D., Trees in polyhedral graphs, Can. J. Math. 18 (1966), 731736.
2. Bondy, J. A. and Entringer, R. C., Longest cycles in 2-connected graphs with prescribed maximum degree, Can. J. Math., to appear.
3. Grünbaum, B. and Motzkin, T. S., Longest simple paths in polyhedral graphs, J. London Math. Soc. 37 (1962), 152160.
4. Grünbaum, B. and Walther, H., Shortness exponents of families of graphs, J. Combinatorial Theory Ser. A. 14 (1973), 364385.
5. Lang, R. and Walther, H., Über längste Kreise in reguldren Graphen, Beitrage zur Graphentheorie (Kolloquium, Manebach, 1967). Teubner, Leipzig (1968), 9198.
6. Tait, P. G., Remarks on colouring of maps, Proc. Royal Soc. Edinburgh Ser. A. 10 (1880), 729.
7. Tutte, W. T., On hamiltoman circuits, J. London Math. Soc. 21 (1946), 98101.
8. Walther, H., Über die Anzahl der Knotenpunkte eines längsten Kreises in planaren, kubischen dreifach knotenzusammenhdngenden Graphen, Studia Sci. Math. Hungar. 2 (1967), 391398.
9. Walther, H., Über Extremalkreise in reguldren Landkarten, Wiss. Z. Techn. Hochsch. Ilmenau 15 (1969), 139142.
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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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