Degree criteria and stability for independent transversals

• Published in: Journal of Graph Theory
• Main Authors: Haxell, Penny, Wdowinski, Ronen
• Other Authors: Natural Sciences and Engineering Research Council of Canada
• Format: Article in Journal/Newspaper
• Language: English
• Published: Wiley 2024
• Subjects: Groenland
• Online Access: http://dx.doi.org/10.1002/jgt.23085; https://onlinelibrary.wiley.com/doi/pdf/10.1002/jgt.23085

Abstract An independent transversal (IT) in a graph with a given vertex partition is an independent set of vertices of (i.e., it induces no edges), that consists of one vertex from each part ( block ) of . Over the years, various criteria have been established that guarantee the existence of an IT, often given in terms of being ‐ thick , meaning all blocks have size at least . One such result, obtained recently by Wanless and Wood, is based on the maximum average block degree . They proved that if then an IT exists. Resolving a problem posed by Groenland, Kaiser, Treffers and Wales (who showed that the ratio 1/4 is best possible), here we give a full characterization of pairs such that the following holds for every : whenever is a graph with maximum degree , and is a ‐thick vertex partition of such that , there exists an IT of with respect to . Our proof makes use of another previously known criterion for the existence of ITs that involve the topological connectedness of the independence complex of graphs, and establishes a general technical theorem on the structure of graphs for which this parameter is bounded above by a known quantity. Our result interpolates between the criterion and the old and frequently applied theorem that if then an IT exists. Using the same approach, we also extend a theorem of Aharoni, Holzman, Howard and Sprüssel, by giving a stability version of the latter result.