L(2,1)-Labeling of the Strong Product of Paths and Cycles

Abstract

AnL(2,1)-labeling of a graphG=(V,E)is a functionffrom the vertex setV(G)to the set of nonnegative integers such that the labels on adjacent vertices differ by at least two and the labels on vertices at distance two differ by at least one. The span offis the difference between the largest and the smallest numbers inf(V). Theλ-number ofG, denoted byλ(G), is the minimum span over allL(2,1)-labelings ofG. We consider theλ-number ofPn⊠Cmand forn≤11theλ-number ofCn⊠Cm. We determineλ-numbers of graphs of interest with the exception of a finite number of graphs and we improve the bounds on theλ-number ofCn⊠Cm,m≥24andn≥26.