L(h, k)-coloring

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 $h, k$ 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, $ρ h,k (G)$ 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 $λ h,k$ -number of $G$ , denoted by $λ h,k (G)$ .

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).

References

^ Chartrand, Gary; Zhang, Ping (2009). "14. Colorings, Distance, and Domination". Chromatic Graph Theory. CRC Press. pp. 397–438.
^ 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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[e01-1] Chartrand, Gary; Zhang, Ping (2009). "14. Colorings, Distance, and Domination". Chromatic Graph Theory. CRC Press. pp. 397–438.

[Tiziana-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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References