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A traditional Nordhaus-Gaddum problem for a graph parameter β is to find a (tight) upper or lower bound on the sum or product of β(G)and β(G¯) (where G¯ denotes the complement of G). An r-decomposition G1,…,Gr of the complete graph Kn is a partition of the edges of Kn among r spanning subgraphs G1,…,Gr. A traditional Nordhaus-Gaddum problem can be viewed as the special case for r=2 of a more general r-part sum or product Nordhaus-Gaddum type problem. We determine the values of the r-part sum and product upper bounds asymptotically as n goes to infinity for the parameters tree-width and its variants largeur d'arborescence, path-width, and proper path-width. We also establish ranges for the lower bounds for these parameters, and ranges for the upper and lower bounds of the r-part Nordhaus-Gaddum type problems for the parameters Hadwiger number, the Colin de Verdière number μ that is used to characterize planarity, and its variants ν and ξ.


This is a pre-print of the article Hogben, Leslie, Jephian C-H. Lin, and Michael Young. "Multi-part Nordhaus-Gaddum type problems for tree-width, Colin de Verdiere type parameters, and Hadwiger number." arXiv preprint arXiv:1604.08817v1 (2016). Posted with permission.

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