L(j,k)- and circular L(j,k)-labellings for the products of complete graphs

Peter Che Bor Lam (), Wensong Lin and Jianzhuan Wu
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Peter Che Bor Lam: Hong Kong Baptist University
Wensong Lin: Hong Kong Baptist University
Jianzhuan Wu: Southeast University

Journal of Combinatorial Optimization, 2007, vol. 14, issue 2, No 12, 219-227

Abstract: Abstract Let j and k be two positive integers with j≥k. An L(j,k)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between labels of any two adjacent vertices is at least j, and the difference between labels of any two vertices that are at distance two apart is at least k. The minimum range of labels over all L(j,k)-labellings of a graph G is called the λ j,k -number of G, denoted by λ j,k (G). A σ(j,k)-circular labelling with span m of a graph G is a function f:V(G)→{0,1,…,m−1} such that |f(u)−f(v)| m ≥j if u and v are adjacent; and |f(u)−f(v)| m ≥k if u and v are at distance two apart, where |x| m =min {|x|,m−|x|}. The minimum m such that there exists a σ(j,k)-circular labelling with span m for G is called the σ j,k -number of G and denoted by σ j,k (G). The λ j,k -numbers of Cartesian products of two complete graphs were determined by Georges, Mauro and Stein ((2000) SIAM J Discret Math 14:28–35). This paper determines the λ j,k -numbers of direct products of two complete graphs and the σ j,k -numbers of direct products and Cartesian products of two complete graphs.

Keywords: λ j; k -number; σ j; k -number; Cartesian product; Direct product (search for similar items in EconPapers)
Date: 2007
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DOI: 10.1007/s10878-007-9057-5

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