Geodesic
Optimal L(h,k)-Labeling of Regular Grids
Discrete mathematics & theoretical computer science, Tome 8 (2006).

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The L(h, k)-labeling is an assignment of non negative integer labels to the nodes of a graph such that 'close' nodes have labels which differ by at least k, and 'very close' nodes have labels which differ by at least h. The span of an L(h,k)-labeling is the difference between the largest and the smallest assigned label. We study L(h, k)-labelings of cellular, squared and hexagonal grids, seeking those with minimum span for each value of k and h ≥ k. The L(h,k)-labeling problem has been intensively studied in some special cases, i.e. when k=0 (vertex coloring), h=k (vertex coloring the square of the graph) and h=2k (radio- or λ -coloring) but no results are known in the general case for regular grids. In this paper, we completely solve the L(h,k)-labeling problem on regular grids, finding exact values of the span for each value of h and k; only in a small interval we provide different upper and lower bounds. 
@article{DMTCS_2006_8_a2,
     author = {Calamoneri, Tiziana},
     title = {Optimal {L(h,k)-Labeling} of {Regular} {Grids}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {8},
     year = {2006},
     doi = {10.46298/dmtcs.361},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.361/}
}
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Calamoneri, Tiziana. Optimal L(h,k)-Labeling of Regular Grids. Discrete mathematics & theoretical computer science, Tome 8 (2006). doi : 10.46298/dmtcs.361. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.361/

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