The Number of Graphs Not Containing $K_{3,3}$ as a Minor

Stefanie Gerke, Omer Giménez, Marc Noy, Andreas Weißl

Abstract


We derive precise asymptotic estimates for the number of labelled graphs not containing $K_{3,3}$ as a minor, and also for those which are edge maximal. Additionally, we establish limit laws for parameters in random $K_{3,3}$-minor-free graphs, like the number of edges. To establish these results, we translate a decomposition for the corresponding graphs into equations for generating functions and use singularity analysis. We also find a precise estimate for the number of graphs not containing the graph $K_{3,3}$ plus an edge as a minor.


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