L(2,1)-coloring explained
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.[1]
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
such that a given graph has an L(2,1)-coloring using color numbers from 0 to
. 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
it is at most
.
[2] Notes and References
- Book: Chartrand . Gary . Zhang . Ping. Ping Zhang (graph theorist) . Gary Chartrand . Chromatic Graph Theory . 2009 . CRC Press . 14. Colorings, Distance, and Domination . 397–438.
- Griggs . Jerrold R. . Yeh . Roger K. . 10.1137/0405048 . 4 . . 1186826 . 586–595 . Labelling graphs with a condition at distance 2 . 5 . 1992.
