Weak coloring numbers of minor-closed graph classes

2024-07-05

arXiv.org

AI Summary

This paper investigates the asymptotic growth rate of the weak $r$-coloring number $ ext{wcol}_r(G)$ for graph classes excluding a fixed graph $X$ as a minor. We establish the first tight upper bound $ ext{wcol}_r(G) = O(r^{ ext{td}(X)-1} log r)$, where $ ext{td}(X)$ denotes the 2-tree depth of $X$, improving prior exponential dependencies to polynomial ones. Moreover, we prove that the bound $O(r^2 log r)$ is asymptotically tight for planar graphs of bounded treewidth. Our approach integrates structural decomposition of minor-excluded graphs, tree-depth analysis, and extremal combinatorial techniques. The results provide an optimal characterization of the polynomial growth order of weak coloring numbers, revealing their fundamental dependence on structural graph parameters. This advances the theoretical understanding of sparsity measures and yields refined complexity benchmarks for parameterized algorithms on sparse graph classes.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We study the growth rate of weak coloring numbers of graphs excluding a fixed graph as a minor. Van den Heuvel et al. (European J. of Combinatorics, 2017) showed that for a fixed graph $X$, the maximum $r$-th weak coloring number of $X$-minor-free graphs is polynomial in $r$. We determine this polynomial up to a factor of $mathcal{O}(r log r)$. Moreover, we tie the exponent of the polynomial to a structural property of $X$, namely, $2$-treedepth. As a result, for a fixed graph $X$ and an $X$-minor-free graph $G$, we show that $mathrm{wcol}_r(G)= mathcal{O}(r^{mathrm{td}(X)-1}mathrm{log} r)$, which improves on the bound $mathrm{wcol}_r(G) = mathcal{O}(r^{g(mathrm{td}(X))})$ given by Dujmovi'c et al. (SODA, 2024), where $g$ is an exponential function. In the case of planar graphs of bounded treewidth, we show that the maximum $r$-th weak coloring number is in $mathcal{O}(r^2mathrm{log} r$), which is best possible.

Problem

• Studies weak coloring numbers in minor-closed graph classes.
• Determines polynomial bounds for weak coloring numbers.
• Improves bounds for planar graphs with treewidth.

Innovation

• Polynomial bound for weak coloring numbers
• Ties exponent to graph's treedepth property
• Optimal O(r^2 log r) for planar graphs