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An equitable partition into branchings in a digraph is a partition of the arcset into branchings such that the sizes of any two branchings differ at most byone. For a digraph whose arc set can be partitioned into $k$ branchings, therealways exists an equitable partition into $k$ branchings. In this paper, wepresent two extensions of equitable partitions into branchings in digraphs:those into matching forests in mixed graphs; and into $b$-branchings indigraphs. For matching forests, Kir\'{a}ly and Yokoi (2022) considered atricriteria equitability based on the sizes of the matching forest, and thematching and branching therein. In contrast to this, we introduce asingle-criterion equitability based on the number of covered vertices, which isplausible in the light of the delta-matroid structure of matching forests.While the existence of this equitable partition can be derived from a lemma inKir\'{a}ly and Yokoi, we present its direct and simpler proof. For$b$-branchings, we define an equitability notion based on the size of the$b$-branching and the indegrees of all vertices, and prove that an equitablepartition always exists. We then derive the integer decomposition property ofthe associated polytopes.

Notes on Equitable Partitions into Matching Forests in Mixed Graphs and into $b$-branchings in Digraphs