CODES AND $L(2,1)$-LABELINGS IN SIERPIŃSKI GRAPHS

Abstract

The $\lambda$-number of a graph $G$ is the minimum value $\lambda$ such that $G$ admits a labeling with labels from $\{0,1,\ldots,\lambda\}$ 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 $n\geq 2$ and any $k \geq 3$, $\lambda(S(n,k)) = 2k$. To obtain the result (perfect) codes in Sierpiński graphs are studied in detail. In particular a new proof of their (essential) uniqueness is obtained.