Improper choosability and Property B
Ross J. Kang
TL;DR
This paper explores the relationship between improper list vertex colourings of graphs and Property B, extending classical results to broader graph classes and list assignment conditions.
Contribution
It introduces new connections between improper list colourings and Property B, generalizing previous findings for multipartite graphs, high minimum degree graphs, and bounded list unions.
Findings
Extended the link between improper list colourings and Property B.
Generalized results to multipartite graphs and graphs with large minimum degree.
Analyzed list assignments with bounded union sizes.
Abstract
A fundamental connection between list vertex colourings of graphs and Property B (also known as hypergraph 2-colourability) was already known to Erd\H{o}s, Rubin and Taylor. In this article, we draw similar connections for improper list colourings. This extends results of Kostochka, Alon, and Kr\'al' and Sgall for, respectively, multipartite graphs, graphs of large minimum degree, and list assignments with bounded list union.
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