The article currently writes:
[The matroid partitioning algorithm's] correctness can be used to prove that a matroid sum is necessarily a matroid.[1][2]
However, I cannot find justification for this sentence in the sources cited. As far as I can tell, neither of the sources explicitly proves that the matroid sum is a matroid, and it is not obvious to me how it follows. Oxley's textbook[3] shows that the matroid sum is a matroid (Theorem 11.3.1) by proving it is induced across a bipartite graph by the direct sum matroid; in turn, the "matroid induced across a bipartite graph by another matroid" is shown to be a matroid using submodular functions (Theorem 11.2.12). Generally the proofs I've found of this fact all use submodular functions rather than the matroid partitioning algorithm.
So, can anyone find a citation proving that the matroid sum is a matroid using the matroid partitioning algorithm, or explain why this is present in the existing sources?
Elestrophe (talk) 19:59, 12 March 2026 (UTC)
