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Graphs and Combinatorics

, Volume 17, Issue 3, pp 473–477 | Cite as

Solution of a Conjecture of Volkmann on the Number of Vertices in Longest Paths and Cycles of Strong Semicomplete Multipartite Digraphs

  • Gregory Gutin
  • Anders Yeo

Abstract.

 A digraph obtained by replacing each edge of a complete multipartite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete multipartite digraph. L. Volkmann conjectured that l≤2c−1, where l (c, respectively) is the number of vertices in a longest path (longest cycle) of a strong semicomplete multipartite digraph. The bound on l is sharp. We settle this conjecture in affirmative.

Keywords

Longe Path Longe Cycle Complete Multipartite Graph Semicomplete Multipartite Digraph 
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Copyright information

© Springer-Verlag Tokyo 2001

Authors and Affiliations

  • Gregory Gutin
    • 1
  • Anders Yeo
    • 2
  1. 1.Department of Mathematics and Statistics, Brunel, The University of West London, Uxbridge, Middlesex, UB8 3PH, UK. e-mail: z.g.gutin@brunel.ac.ukGB
  2. 2.Department of Mathematics and Statistics, University of Victoria, P.O. Box 3045, Victoria B.C., Canada V8W 3P4. e-mail: yeo@Math.UVic.caCA

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