Defective coloring is perfect for minors
Chun-Hung Liu

TL;DR
This paper proves that the defective chromatic number of any minor-closed graph family matches the simple lower bound, confirming a conjecture and providing optimal minors for infinite graphs with bounded degree partitions.
Contribution
It confirms a conjecture that the defective chromatic number equals the lower bound for all minor-closed families, resolving a key open problem.
Findings
Defective chromatic number equals the simple lower bound for minor-closed families.
Provides the optimal list of unavoidable minors for infinite graphs with bounded degree partitions.
Establishes a linear relation between clustered chromatic number and forbidden minors' tree-depth.
Abstract
The defective chromatic number of a graph class is the infimum such that there exists an integer such that every graph in this class can be partitioned into at most induced subgraphs with maximum degree at most . Finding the defective chromatic number is a fundamental graph partitioning problem and received attention recently partially due to Hadwiger's conjecture about coloring minor-closed families. In this paper, we prove that the defective chromatic number of any minor-closed family equals the simple lower bound obtained by the standard construction, confirming a conjecture of Ossona de Mendez, Oum, and Wood. This result provides the optimal list of unavoidable finite minors for infinite graphs that cannot be partitioned into a fixed finite number of induced subgraphs with uniformly bounded maximum degree. As corollaries about clustered coloring, we obtain a linear…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph Labeling and Dimension Problems
