The graph genus problem is NP-complete
Journal of Algorithms
Planar Separators and the Euclidean Norm
SIGAL '90 Proceedings of the International Symposium on Algorithms
Edge Separators for Graphs of Bounded Genus with Applications
WG '91 Proceedings of the 17th International Workshop
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Given a graph G=(V,E) with n vertices, m edges and maximum vertex degree @D, the load distribution of a coloring @f:V-{red, blue} is a pair d"@f=(r"@f,b"@f), where r"@f is the number of edges with at least one end-vertex colored red and b"@f is the number of edges with at least one end-vertex colored blue. Our aim is to find a coloring @f such that the (maximum) load, l"@f:=1m@?max{r"@f,b"@f}, is minimized. This problems arises in Wavelength Division Multiplexing (WDM), the technology currently in use for building optical communication networks. After proving that the general problem is NP-hard we give a polynomial time algorithm for optimal colorings of trees and show that the optimal load is at most 1/2+(@D/m)log"2n. For graphs with genus g0, we show that a coloring with load OPT(1+o(1)) can be computed in O(n+glogn)-time, if the maximum degree satisfies @D=o(m^2ng) and an embedding is given. In the general situation we show that a coloring with load at most 34+O(@D/m) can be found by analyzing a random coloring with Chebychev's inequality. This bound describes the ''typical'' situation: in the random graph model G(n,m) we prove that for almost all graphs, the optimal load is at least 34-n/m. Finally, we state some conjectures on how our results generalize to k-colorings for k2.