Distance-two labelings of digraphs

  • Authors:

  • Gerard J. Chang;Jer-Jeong Chen;David Kuo;Sheng-Chyang Liaw

  • Affiliations:

  • Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan and Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan and National Center for ...;Department of Marketing and Logistics, China University of Technology, Taipei 116, Taiwan and National Center for Theoretical Sciences, Taipei Office;Department of Applied Mathematics, National Dong Hwa University, Hualien 974, Taiwan and National Center for Theoretical Sciences, Taipei Office;Department of Mathematics, National Central University, Chungli 32054, Taiwan and National Center for Theoretical Sciences, Taipei Office

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2007

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Abstract

For positive integers j=k, an L(j,k)-labeling of a digraph D is a function f from V(D) into the set of nonnegative integers such that |f(x)-f(y)|=j if x is adjacent to y in D and |f(x)-f(y)|=k if x is of distance two to y in D. Elements of the image of f are called labels. The L(j,k)-labeling problem is to determine the @l@?"j","k-number @l@?"j","k(D) of a digraph D, which is the minimum of the maximum label used in an L(j,k)-labeling of D. This paper studies @l@?"j","k-numbers of digraphs. In particular, we determine @l@?"j","k-numbers of digraphs whose longest dipath is of length at most 2, and @l@?"j","k-numbers of ditrees having dipaths of length 4. We also give bounds for @l@?"j","k-numbers of bipartite digraphs whose longest dipath is of length 3. Finally, we present a linear-time algorithm for determining @l@?"j","1-numbers of ditrees whose longest dipath is of length 3.