The L(2, 1)-Labeling Problem on Oriented Regular Grids

Abstract

The L(2, 1)-labeling of a digraph G is a function f from the node set of $G$ to the set of all nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $x$ and $y$ are at distance 1, and $f(x)=f(y)$ if $x$ and $y$ are at distance 2, where the distance from vertex $x$ to vertex $y$ is the length of a shortest dipath from $x$ to $y$. The minimum of the maximum used label over all $L(2,1)$-labelings of $G$ is called $\lambda (G)$. In this paper we study the L(2, 1)-labeling problem on squared, triangular and hexagonal grids and for them we compute the exact values of $\lambda$.

The content of this paper has been presented to ICTCS 2009, 28-30 September, Cremona, Italy. This updated version is a longer and more complete version of the first submission (from 10 to 13 pages, from 5 to 7 figures) and a wrong figure has been corrected