On the number of graphs not containing $K_{3,3}$ as a minor

Details On the number of graphs not containing $K_{3,3}$ as a minor On the number of graphs not containing $K_{3,3}$ as a minor

2008-03-31

arxiv.org/abs/0803.4418v2

Mathematics

Combinatorics

17 pages

Scientific paper

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 expected number of edges. To establish these results, we translate a decomposition for the corresponding graph class 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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