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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 ${\displaystyle n}$ such that a given graph has an L(2,1)-coloring using color numbers from 0 to ${\displaystyle n}$. 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${\displaystyle \Delta }$ it is at most ${\displaystyle \Delta ^{2}+2\Delta }$.

References

1. Chartrand, Gary; Zhang, Ping (2009). "14. Colorings, Distance, and Domination". Chromatic Graph Theory. CRC Press. pp. 397–438.
2. Griggs, Jerrold R.; Yeh, Roger K. (1992). "Labelling graphs with a condition at distance 2". SIAM Journal on Discrete Mathematics. 5 (4): 586–595. doi:10.1137/0405048. MR 1186826.

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