On the Impossibility of Decomposing Binary Matroids

We show that there exist $k$-colorable matroids that are not $(b,c)$-decomposable when $b$ and $c$ are constants. A matroid is $(b,c)$-decomposable, if its ground set of elements can be partitioned into sets $X_1, X_2, \ldots, X_l$ with the following two properties. Each set $X_i$ has size at most $ck$. Moreover, for all sets $Y$ such that $|Y \cap X_i| \leq 1$ it is the case that $Y$ is $b$-colorable. A $(b,c)$-decomposition is a strict generalization of a partition decomposition and, thus, our result refutes a conjecture from arXiv:1911.10485v2 .