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On defining numbers of circular complete graphs

Daneshgar, A. (Amir) ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1016/j.disc.2006.07.002
  3. Abstract:
  4. Let d(σ)d(σ) stand for the defining number of the colouring σσ. In this paper we consider View the MathML sourcedmin=minγd(γ) and View the MathML sourcedmax=maxγd(γ) for the onto χχ-colourings γγ of the circular complete graph Kn,dKn,d. In this regard we obtain a lower bound for dmin(Kn,d)dmin(Kn,d) and we also prove that this parameter is asymptotically equal to χ-1χ-1. Also, we show that when χ⩾4χ⩾4 and s≠0s≠0 then dmax(Kχd-s,d)=χ+2s-3dmax(Kχd-s,d)=χ+2s-3, and, moreover, we prove an inequality relating this parameter to the circular chromatic number for any graph G
  5. Keywords:
  6. Defining set ; Graph colouring ; Circular colouring
  7. Source: Discrete Mathematics ; Volume 307, Issue 2, 28 January 2007, Pages 173–180
  8. URL: http://www.sciencedirect.com/science/article/pii/S0012365X0600481X