L(2,1)-coloring

L(2,1) coloring of the cycle C₆

L(2, 1)-coloring or L(2,1)-labeling is a particular case of L(h, k)-coloring. In an L(2, 1)-coloring of a graph, G, the vertices of G are assigned color numbers in such a way that adjacent vertices get labels that differ by at least two, and the vertices that are at a distance of two from each other get labels that differ by at least one.

An L(2,1)-coloring is a proper coloring, since adjacent vertices are assigned distinct colors. However, rather than counting the number of distinct colors used in an L(2,1)-coloring, research has centered on the L(2,1)-labeling number, the smallest integer $n$ such that a given graph has an L(2,1)-coloring using color numbers from 0 to $n$. The L(2,1)-coloring problem was introduced in 1992 by Jerrold Griggs and Roger Yeh, motivated by channel allocation schemes for radio communication. They proved that for cycles, such as the 6-cycle shown, the L(2,1)-labeling number is four, and that for graphs of degree$\Delta$ it is at most $\Delta ^{2}+2\Delta$.