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Hamiltonicity in connected regular graphs

Profile image of Suil OhSuil Oh

2013, Information Processing Letters

https://doi.org/10.1016/J.IPL.2013.08.005
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Abstract

In 1980, Jackson proved that every 2-connected k-regular graph with at most 3k vertices is Hamiltonian. This result has been extended in several papers. In this note, we determine the minimum number of vertices in a connected k-regular graph that is not Hamiltonian, and we also solve the analogous problem for Hamiltonian paths. Further, we characterize the smallest connected k-regular graphs without a Hamiltonian cycle.

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References (6)

  1. F. Hilbig, Kantenstrukturen in nichthamiltonschen Graphen. Ph.D. Thesis, Technische Universität Berlin (1986).
  2. B. Jackson, Hamilton cycles in regular 2-connected graphs, J. Combin. Theorey B 29 (1980), 27-46.
  3. B. Jackson, Cycles through vertices of large maximum degree, J. Graph Theory 19 (1995), 157-168.
  4. Suil O, D.B. West, Balloons, cut-edges, matchings, and total domination in regular graphs of odd degree. Journal of Graph Theory 64 (2010) 116-131.
  5. Suil O, D.B. West, Matchings, and edge-connectivity in regular graphs. European J. Combinatorics 32 (2011) 324-329.
  6. D.B. West, Introduction to Graph Theory, Prentice Hall, INC., Upper Sadle River, NJ, 2001.

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