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


 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.


Stated Problem Local Irregularity Infinite Class Semicomplete Multipartite Digraph Global Irregularity 
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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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