Multi-part Nordhaus-Gaddum type problems for tree-width, Colin de Verdi\`ere type parameters, and Hadwiger number

TL;DR

This paper investigates multi-part Nordhaus-Gaddum problems for various graph parameters, providing asymptotic bounds for sums and products across multiple graph decompositions, extending classical two-graph analyses.

Contribution

It generalizes Nordhaus-Gaddum problems to multiple parts and determines asymptotic bounds for several graph parameters, including tree-width and Hadwiger number.

Findings

Asymptotic upper bounds for r-part sums and products of tree-width and variants.

Ranges for lower bounds of these parameters in r-part decompositions.

Bounds for Hadwiger number and Colin de Verdière number in multi-part settings.

Abstract

A traditional Nordhaus-Gaddum problem for a graph parameter $\beta$ is to find a (tight) upper or lower bound on the sum or product of $\beta(G)$ and $\beta(\bar{G})$ (where $\bar{G}$ denotes the complement of $G$). An $r$-decomposition $G_1,\dots,G_r$ of the complete graph $K_n$ is a partition of the edges of $K_n$ among $r$ spanning subgraphs $G_1,\dots,G_r$. 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 Verdi\`ere number $\mu$ that is used to characterize planarity, and its variants $\nu$ and $\xi$.