Notes on Equitable Partitions into Matching Forests in Mixed Graphs and into $b$-branchings in Digraphs
Abstract
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 branchings, there always exists an equitable partition into branchings. In this paper, we present two extensions of equitable partitions into branchings in digraphs: those into matching forests in mixed graphs; and into -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 -branchings, we define an equitability notion based on the size of the -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.
Keywords
Cite
@article{arxiv.2003.10774,
title = {Notes on Equitable Partitions into Matching Forests in Mixed Graphs and into $b$-branchings in Digraphs},
author = {Kenjiro Takazawa},
journal= {arXiv preprint arXiv:2003.10774},
year = {2023}
}