The graphicity of the union of graphic matroids

Abstract

There is a conjecture that if the union (also called sum) of graphic matroids is not graphic then it is nonbinary (Recski, 1982). 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 given for the other matroid to ensure the graphicity of the union. These conditions can be checked in polynomial time. The proofs imply that the above conjecture holds for these cases.