Distance-two labellings of Hamming graphs

  • Authors:

  • Gerard J. Chang;Changhong Lu;Sanming Zhou

  • 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 Mathematics, East China Normal University, Shanghai 200062, PR China and Institute of Theoretical Computing, East China Normal University, Shanghai 200062, PR China;Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2009

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Abstract

Let j=k=0 be integers. An @?-L(j,k)-labelling of a graph G=(V,E) is a mapping @f:V-{0,1,2,...,@?} such that |@f(u)-@f(v)|=j if u,v are adjacent and |@f(u)-@f(v)|=k if they are distance two apart. Let @l"j","k(G) be the smallest integer @? such that G admits an @?-L(j,k)-labelling. Define @l@?"j","k(G) to be the smallest @? if G admits an @?-L(j,k)-labelling with @f(V)={0,1,2,...,@?} and ~ otherwise. An @?-cyclic L(j,k)-labelling is a mapping @f:V-Z"@? such that |@f(u)-@f(v)|"@?=j if u,v are adjacent and |@f(u)-@f(v)|"@?=k if they are distance two apart, where |x|"@?=min{x,@?-x} for x between 0 and @?. Let @s"j","k(G) be the smallest @?-1 of such a labelling, and define @s@?"j","k(G) similarly to @l@?"j","k(G). We determine @l"2","0, @l@?"2","0, @s"2","0 and @s@?"2","0 for all Hamming graphs K"q"""1@?K"q"""2@?...@?K"q"""d (d=2, q"1=q"2=...=q"d=2) and give optimal labellings, with the only exception being 2q@?@s@?"2","0(K"q@?K"q)@?2q+1 for q=4. We also prove the following ''sandwich theorem'': If q"1 is sufficiently large then @l"2","1(G)=@l@?"2","1(G)=@s@?"2","1(G)=@s"2","1(G)=@l"1","1(G)=@l@?"1","1(G)=@s@?"1","1(G)=@s"1","1(G)=q"1q"2-1 for any graph G between K"q"""1@?K"q"""2 and K"q"""1@?K"q"""2@?...@?K"q"""d, and moreover we give a labelling which is optimal for these eight invariants simultaneously.