Cooperative colorings of hypergraphs
Xuqing Bai, Bi Li, Weichan Liu, Xin Zhang

TL;DR
This paper introduces the concept of cooperative colorings in hypergraphs, determining their minimal number for specific classes like cycles and paths, and establishing bounds for k-partite hypergraphs with large degrees.
Contribution
It provides exact cooperative chromatic numbers for certain hypergraph classes and bounds for k-partite hypergraphs with high degree, using a new set system partition theorem.
Findings
Exact cooperative chromatic number of 2 for specific hypergraph classes.
Bounds on cooperative chromatic number for k-partite hypergraphs with large degree.
Introduction of a new set system partition theorem.
Abstract
Given a class of hypergraphs with the same vertex set , a cooperative coloring of them is a partition of in such a way that each is an independent set in for . The cooperative chromatic number of a class is the smallest number of hypergraphs from that always possess a cooperative coloring. For the classes of -uniform tight cycles, -uniform loose cycles, -uniform tight paths, and -uniform loose paths, we find that their cooperative chromatic numbers are all exactly two utilizing a new proved set system partition theorem, which also has its independent interests and offers a broader perspective. For the class of -partite -uniform hypergraphs with sufficient large maximum degree , we prove that its cooperative chromatic number has…
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Taxonomy
TopicsGraph Labeling and Dimension Problems
