Matroid union — Graphic? Binary? Neither?

Abstract

There is a conjecture that if the union (also called sum) of graphic matroids is not graphic then it is nonbinary. Some special cases have been proved only, for example if several copies of the same graphic matroid are given. If there are two matroids and the first one can either be represented by a graph with two points, or is the direct sum of a circuit and some loops, then a necessary and sufficient condition is known for the other matroid to ensure the graphicity of the union and the above conjecture holds for these cases. We prove the sufficiency of this condition for the graphicity of the union of two arbitrary graphic matroids. Then we present a weaker necessary condition which is of similar character. Finally we suggest a more general framework of the study of such questions by introducing matroid classes formed by those graphic (or arbitrary) matroids whose union with any graphic (or arbitrary) matroid is graphic (or either graphic or nonbinary).