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Graphs and Combinatorics

, Volume 11, Issue 3, pp 221–231 | Cite as

Hamiltonicity in balancedk-partite graphs

  • Guantao Chen
  • Ralph J. Faudree
  • Ronald J. Gould
  • Michael S. Jacobson
  • Linda Lesniak
Original Papers

Abstract

One of the earliest results about hamiltonian graphs was given by Dirac. He showed that if a graphG has orderp and minimum degree at least\(\frac{p}{2}\) thenG is hamiltonian. Moon and Moser showed that a balanced bipartite graph (the two partite sets have the same order)G has orderp and minimum degree more than\(\frac{p}{4}\) thenG is hamiltonian. In this paper, their idea is generalized tok-partite graphs and the following result is obtained: LetG be a balancedk-partite graph with orderp = kn. If the minimum degree
$$\delta (G) > \left\{ {\begin{array}{*{20}c} {\left( {\frac{k}{2} - \frac{1}{{k + 1}}} \right)n if k is odd } \\ {\left( {\frac{k}{2} - \frac{2}{{k + 2}}} \right)n if k is even} \\ \end{array} } \right.$$
thenG is hamiltonian. The result is best possible.

Keywords

Early Result Bipartite Graph Minimum Degree Hamiltonian Graph Balance Bipartite Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Dirac, G.: Some theorems on abstract graphs. Proc. London Math Soc.2, 69–81 (1952)Google Scholar
  2. 2.
    Moon, J., Moser L.: On hamiltonian bipartite graphs. Israel J. Math1, 163–165 (1963)Google Scholar
  3. 3.
    Ore, O.: Note on hamiltonian circuits. Amer. Math Monthly67, 55 (1960)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Guantao Chen
    • 1
  • Ralph J. Faudree
    • 2
  • Ronald J. Gould
    • 3
  • Michael S. Jacobson
    • 4
  • Linda Lesniak
    • 5
  1. 1.Department of MathematicsNorth Dakota State UniversityFargoUSA
  2. 2.Department of Mathematical SciencesUniversity of MemphisMemphisUSA
  3. 3.Department of Math. & Comp. Sci.Emory UniversityAtlantaUSA
  4. 4.Department of MathematicsUniversity of LouisvilleLouisvilleUSA
  5. 5.Department of Math. & Comp. Sci.Drew UniversityMadisonUSA

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