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Graphs and Combinatorics

, Volume 15, Issue 4, pp 481–493 | Cite as

How Close to Regular Must a Semicomplete Multipartite Digraph Be to Secure Hamiltonicity?

  • Anders Yeo

Abstract.

 Let D be a semicomplete multipartite digraph, with partite sets V1, V2,…, Vc, such that |V1|≤|V2|≤…≤|Vc|. Define f(D)=|V(D)|−3|Vc|+1 and \(\). We define the irregularity i(D) of D to be max|d+(x)−d(y)| over all vertices x and y of D (possibly x=y). We define the local irregularity il(D) of D to be max|d+(x)−d(x)| over all vertices x of D and we define the global irregularity of D to be ig(D)=max{d+(x),d(x) : xV(D)}−min{d+(y),d(y) : yV(D)}. In this paper we show that if ig(D)≤g(D) or if il(D)≤min{f(D), g(D)} then D is Hamiltonian. We furthermore show how this implies a theorem which generalizes two results by Volkmann and solves a stated problem and a conjecture from [6]. Our result also gives support to the conjecture from [6] that all diregular c-partite tournaments (c≥4) are pancyclic, and it is used in [9], which proves this conjecture for all c≥5. Finally we show that our result in some sense is best possible, by giving an infinite class of non-Hamiltonian semicomplete multipartite digraphs, D, with ig(D)=i(D)=il(D)=g(D)+½≤f(D)+1.

Keywords

Stated Problem Local Irregularity Infinite Class Semicomplete Multipartite Digraph Global Irregularity 
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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Copyright information

© Springer-Verlag Tokyo 1999

Authors and Affiliations

  • Anders Yeo
    • 1
  1. 1.Department of Mathematics and Computer Science, Odense University, DK-5230 Odense, Denmark. e-mail: gyeo@imada.ou.dkDK

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