Note: L(2,1)-Labelings on the composition of n graphs

  • Authors:

  • Zhendong Shao;Roberto Solis-Oba

  • Affiliations:

  • Department of Computer Science, The University of Western Ontario, London, ON, Canada;Department of Computer Science, The University of Western Ontario, London, ON, Canada

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2010

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Abstract

An L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)-f(y)|=2 if d(x,y)=1 and |f(x)-f(y)|=1 if d(x,y)=2, where d(x,y) denotes the distance between x and y in G. The L(2,1)-labeling number @l(G) of G is the smallest number k such that G has an L(2,1)-labeling with max{f(v):v@?V(G)}=k. Griggs and Yeh conjectured that @l(G)@?@D^2 for any simple graph with maximum degree @D=2. In this article, a problem in the proof of a theorem in Shao and Yeh (2005) [19] is addressed and the graph formed by the composition of n graphs is studied. We obtain bounds for the L(2,1)-labeling number for graphs of this type that is much better than what Griggs and Yeh conjectured for general graphs. As a corollary, the present bound is better than the result of Shiu et al. (2008) [21] for the composition of two graphs G"1[G"2] if @n"2