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

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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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Brunel, The University of West London, Uxbridge, Middlesex, UB8 3PH, UK. e-mail: z.g.gutin@brunel.ac.uk, , , , , , GB

    Gregory Gutin

  2. Department of Mathematics and Statistics, University of Victoria, P.O. Box 3045, Victoria B.C., Canada V8W 3P4. e-mail: yeo@Math.UVic.ca, , , , , , CA

    Anders Yeo

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  1. Gregory Gutin
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  2. Anders Yeo
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Received: October 26, 1998¶Final version received: August 16, 1999

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Gutin, G., Yeo, A. Solution of a Conjecture of Volkmann on the Number of Vertices in Longest Paths and Cycles of Strong Semicomplete Multipartite Digraphs. Graphs Comb 17, 473–477 (2001). https://doi.org/10.1007/s003730170022

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  • DOI: https://doi.org/10.1007/s003730170022

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