In graph theory, a L(h, k)-labelling, L(h, k)-coloring or sometimes L(p, q)-coloring is a (proper) vertex coloring in which every pair of adjacent vertices has color numbers that differ by at least h, and any nodes connected by a 2 length path have their colors differ by at least k.[1] The parameters, h and k are understood to be non-negative integers.

The problem originated from a channel assignment problem in radio networks. The span of an L(h, k)-labelling, ρh,k(G) is the difference between the largest and the smallest assigned frequency. The goal of the L(h, k)-labelling problem is usually to find a labelling with minimum span.[2] For a given graph, the minimum span over all possible labelling functions is the λh,k-number of G, denoted by λh,k(G).

When h=1 and k=0, it is the usual (proper) vertex coloring.

There is a very large number of articles concerning L(h, k)-labelling, with different h and k parameters and different classes of graphs.

In some variants, the goal is to minimize the number of used colors (the order).

See also

References

  1. ↑ Chartrand, Gary; Zhang, Ping (2009). "14. Colorings, Distance, and Domination". Chromatic Graph Theory. CRC Press. pp. 397–438.
  2. ↑ Tiziana, Calamoneri (2006), "The L(h, k)-Labelling Problem: A Survey and Annotated Bibliography", Comput. J., 49 (5): 585–608, doi:10.1093/comjnl/bxl018


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