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

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

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An equitable partition into branchings in a digraph is a partition of the arc set into branchings such that the sizes of any two branchings differ at most by one.For a digraph whose arc set can be partitioned into $k$ branchings, there always exists an equitable partition into $k$ branchings.In this paper, we present two extensions of equitable partitions into branchings in digraphs: those into matching forests in mixed graphs; and apunisw2 into $b$-branchings in digraphs.For matching forests, Kir'{a}ly and Yokoi (2022) considered a tricriteria equitability based on the sizes of the matching forest, and the matching and branching therein.In contrast to this, we introduce a single-criterion equitability based on the number of covered vertices, which is plausible in the light of the delta-matroid structure of matching forests.

While the existence of this equitable partition can be derived from a lemma in Kir'{a}ly and Yokoi, we present its direct and simpler proof.For $b$-branchings, we define tillman 750m an equitability notion based on the size of the $b$-branching and the indegrees of all vertices, and prove that an equitable partition always exists.We then derive the integer decomposition property of the associated polytopes.