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]
References
edit- ^ Chartrand, Gary; Zhang, Ping (2009). "14. Colorings, Distance, and Domination". Chromatic Graph Theory. CRC Press. pp. 397–438.397-438&rft.pub=CRC Press&rft.date=2009&rft.aulast=Chartrand&rft.aufirst=Gary&rft.au=Zhang, Ping&rfr_id=info:sid/en.wikipedia.org:L(2,1)-coloring" class="Z3988">
- ^ 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 1186826.586-595&rft.date=1992&rft_id=info:doi/10.1137/0405048&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=1186826#id-name=MR&rft.aulast=Griggs&rft.aufirst=Jerrold R.&rft.au=Yeh, Roger K.&rfr_id=info:sid/en.wikipedia.org:L(2,1)-coloring" class="Z3988">