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Atlantic Graph Theory Seminar: Sandra Kingan (Brooklyn College and Graduate Center, CUNY)

December 8, 2021 @ 3:30 pm - 4:30 pm

I will begin by giving a general overview of what it means to find monarchs for excluded minor classes of graphs and matroids. In a paper that appeared in 2018, I used the Strong Splitter Theorem to give a short proof of Oxley’s result that the class of binary matroids with no 4-wheel minor consists of a few small matroids and an infinite family of maximal 3-connected rank r matroids known as the binary spikes. Such a family is called a monarch for the excluded minor class. This proof essentially comes down to finding the monarchs for non-regular matroids with no minors isomorphic to a 9-element rank 4 matroid known as P9 or its dual P*9. In a paper that appeared this year (Australasian Journal of Combinatorics, 79(3), 302–326), I was able to strengthen the result by characterizing the class of binary non-regular matroids with no minor isomorphic to just P*9. The only members of this class are the rank 3 and 4 binary projective geometries, a 16-element rank 5 matroid, and two monarchs: the rank r binary spikes with 2r+1 elements mentioned earlier and another infinite family with 4r−5 elements. As a consequence, a simple binary matroid of rank at least 6 with no P*9-minor has size at most r(r+1)/2 and this bound is attained by the rank r complete graph. This is one of few excluded minor classes for which the members are so precisely determined.


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December 8, 2021
3:30 pm - 4:30 pm
Event Category:


Zoom seminar