Codes and L(2,1)-labelings in sierpiński graphs

Abstract

The λ-number of a graph G is the minimum value λ such that G admits a labeling with labels from {0, 1, . . ., λ} where vertices at distance two get different labels and adjacent vertices get labels that are at least two apart. Sierpiński graphs S(n, k) generalize the Tower of Hanoi graphs - the graph S(n, 3) is isomorphic to the graph of the Tower of Hanoi with n disks. It is proved that for any n ≥ 2 and any k ≥ 3, λ(S(n, k)) = 2k. To obtain the result (perfect) codes in Sierpiński graphs are studied in detail. In particular a new proof of their (essential) uniqueness is obtained.