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

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 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 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.


Introduction
Partitioning a finite set into its subsets with certain combinatorial structure is a fundamental topic in the fields of combinatorial optimization, discrete mathematics, and graph theory. The most typical partitioning problem is graph coloring, which amounts to partitioning the vertex set of a graph into stable sets. In particular, equitable coloring, in which the numbers of vertices in any two stable sets differ at most by one, has attracted researchers' interest since the famous conjecture of Erdős [14] on the existence of an equitable coloring with ∆ + 1 colors in a graph with maximum degree ∆, which was later proved by Hajnal and Szemerédi [21].
Equitable edge-coloring has been mainly considered in bipartite graphs: a bipartite graph with maximum degree ∆ admits an equitable edge-coloring with k colors for every k ≥ ∆ [8,9,11,15,26].
Extension of equitable edge-coloring in bipartite graphs to equitable partition of the common ground set of two matroids into common independent sets has been a challenging topic in the literature [7,16].
One successful example is equitable partition into branchings in digraphs. Let (V, A) denote a digraph with vertex set V and arc set A. In a digraph (V, A), an arc subset B ⊆ A is a branching if every vertex has at most one incoming arc in B and B includes no directed cycle. The following theorem is derived from Edmonds' disjoint branchings theorem [13]. For a real number x, let ⌊x⌋ and ⌈x⌉ denote the maximum integer that is not greater than x and the minimum integer that is not less than x, respectively.
Theorem 1 (see Schrijver [29,Theorem 53.3]). In a digraph D = (V, A), if A can be partitioned into k branchings, then A can be partitioned into k branchings each of which has size ⌊|A|/k⌋ or ⌈|A|/k⌉.
The aim of this paper is to extend of Theorem 1 into two generalizations of branchings: matching forests [18,19,20] and b-branchings [22]. An important feature is that, due to their structures, defining the equitability of matching forests and b-branchings is not a trivial task, as explained below.

Matching Forests
The concept of matching forests was introduced by Giles [18,19,20]. A mixed graph G = (V, E, A) consists of the set V of vertices, the set E of undirected edges, and the set A of directed edges (arcs). We say that an undirected edge in E covers a vertex v ∈ V if v is one of the endpoints of the undirected edge, and a directed edge in A covers v if v is the head of the directed edge. A subset of edges F ⊆ E ∪ A is a matching forest if the underlying edge set of F is a forest and each vertex is covered by at most one edge in F . It is straightforward to see that matching forests offer a common generalization of matchings in undirected graphs and branchings in digraphs: if F ⊆ E ∪ A is a matching forest, then F ∩ E is a matching and F ∩ A is a branching.
An equivalent definition of matching forests can be given in the following way. For a subset of undirected edges M ⊆ E, let ∂M ⊆ V denote the set of vertices covered by at least one edge in M . For a subset of directed edges B ⊆ A, let ∂B ⊆ V denote the set of vertices covered by at least one arc in B (i) .
That is, ∂F is the set of the vertices covered by F . Now F ⊆ E ∪ A is a matching forest if M := F ∩ E is a matching, B := F ∩ A is a branching, and ∂M ∩ ∂B = ∅.
Previous work on matching forests includes polynomial algorithms with polyhedral description [18,19,20], total dual integrality of the description [28], a Vizing-type theorem on the number of matching forests partitioning the edge set E ∪ A [23], and reduction to linear matroid parity [29]. More recently, Takazawa [31] showed that the sets of vertices covered by the matching forests form a delta-matroid [3,5,10]. For two sets X and Y , let X△Y denote their symmetric difference, i.e., X△Y = (X \ Y ) ∪ (Y \ X). For a finite set V and its subset family F ⊆ 2 V , the set system (V, F ) is a delta-matroid if it satisfies the following exchange property: Then, the set system (V, F G ) is a delta-matroid.
(i) We believe that this notation causes no confusion on the direction of the arcs, since we never refer to the set of the tails of the arcs in this paper.
Theorem 2 commonly extends matching delta-matroids [4,5] and matroids defined from branchings (see [27,30]), and provides some explanation of the tractability of matching forests as well as the polyhedral results mentioned above. The most recent work on matching forests is due to Király and Yokoi [24], which discusses equitable partition into matching forests. They considered equitability based on the sizes of F , F ∩ E, and F ∩ A, and proved the following two theorems. Let Z ++ denote the set of the positive integers. For k ∈ Z ++ , let [k] denote the set of positive integers not greater than k, i.e., [k] = {1, . . . , k}.
Theorem 3 (Király and Yokoi [24]). Let G = (V, E, A) be a mixed graph and k ∈ Z ++ . If E ∪ A can be partitioned into k matching forests, then E ∪ A can be partitioned into k matching forests F 1 , . . . , F k such that Theorem 4 (Király and Yokoi [24]). Let G = (V, E, A) be a mixed graph and k ∈ Z ++ . If E ∪ A can be partitioned into k matching forests, then E ∪ A can be partitioned into k matching forests F 1 , . . . , F k such that Moreover, Király and Yokoi [24] showed that both of Theorems 3 and 4 are best possible with this tricriteria equitability, by presenting examples in which (1) and (2) cannot be improved. That is, while attaining the best possible results, Theorems 3 and 4 mean that these three criteria cannot be optimized at the same time, which demonstrates a sort of obscurity of matching forests.
In this paper, we introduce a new equitability, which builds upon a single criterion defined by the number of the covered vertices. Namely, our equitable-partition theorem is described as follows.
Theorem 5. Let G = (V, E, A) be a mixed graph and k ∈ Z ++ . If E ∪ A can be partitioned into k matching forests, then E ∪ A can be partitioned into k matching forests F 1 , . . . , F k such that We remark that this criterion of equitability is plausible in the light of the delta-matroid structure of matching forests (Theorem 2), focusing on the covered vertices rather than the edge set. Theorem 5 contrasts with Theorems 3 and 4 in that the value two in the right-hand side of (3) is tight: consider the case where G = (V, E, A) consists of an odd number of undirected edges forming a path and no directed edges, and k = 2.
We also remark that Theorem 5 can indeed be derived from Lemma 4.2 in [24], as shown in Appendix A. However, the proof of this lemma involves some careful case-by-case analysis, requiring ten types of alternating paths. In contrast to this, we present a direct and simpler proof for Theorem 5, including only one type of alternating paths, by extending the argument in the proof for the exchangeability of matching forests by Schrijver [28,Theorem 2]. Schrijver used this property to prove the total dual integrality of the linear system describing the matching forest polytope presented by Giles [19], and the delta-matroid structure of matching forests [31] is also derived from an in-depth analysis of the proof for the exchangeability.

b-branchings
We next address equitable partition of a digraph into b-branchings, introduced by Kakimura, Kamiyama, and Takazawa [22]. Let Z V ++ denote the set of the |V |-dimensional vectors of which each coordinate corresponds to an element in V and each component is a positive integer.
Note that the branchings is a special case of b-branchings where b(v) = 1 for every v ∈ V . That is, bbranchings provide a generalization of branchings in which the indegree bound of each vertex v ∈ V can be an arbitrary positive integer b(v) (Condition (4)). Together with Condition (5), it yields a reasonable generalization of branchings admitting extensions of several fundamental results on branchings, such as a multi-phase greedy algorithm [2,6,12,17], a packing theorem [13], and the integer decomposition property of the corresponding polytope [1]. The packing theorem on b-branchings leads to a necessary and sufficient condition for the arc set A to be partitionable into k b-branchings [22]. In this paper, we prove that an equitable partition into k b-branchings always exists, provided that any partition into k b-branchings exists.
. . , B k satisfying the following: for every v ∈ V , Theorem 6 exactly coincides with Theorem 1. A new feature is that our definition of equitability of b-branchings is twofold: the number of arcs in any two b-branchings differ at most one (Condition 1); and the indegrees of each vertex with respect to any two b-branchings differ at most one (Condition 2). Theorem 6 means that the optimality of these |V | + 1 criteria can be attained at the same time, which suggests some good structure of b-branchings.
One consequence of Theorem 6 is the integer decomposition property of the convex hull of b-branchings of fixed size and indegrees. For a polytope P and a positive integer κ, define κP = {x | ∃x ′ ∈ P , x = κx ′ }. A polytope P has the integer decomposition property if, for every κ ∈ Z ++ and every integer vector x ∈ κP , there exist κ integer vectors x 1 , . . . , x κ such that x = x 1 + · · · + x κ .
For branchings, Baum and Trotter [1] showed that the branching polytope has the integer decomposition property. Moreover, McDiarmid [25] proved the integer decomposition property of the convex hull of branchings of fixed size ℓ. For b-branchings, the integer decomposition property of the b-branching polytope is proved in [22]: Theorem 7 (Kakimura, Kamiyama and Takazawa [22]). Let D = (V, A) be a digraph and b ∈ Z V ++ . Then, the b-branching polytope has the integer decomposition property.
In this paper, we derive the integer decomposition property of the convex hull of b-branchings of fixed size and indegrees from Theorems 6 and 7. Let Z + denote the set of nonnegative integers, and define Z V ′ + for V ′ ⊆ V in a similar way to the definition of Z V ++ .
Then, the convex hull of incidence vectors of the b-branchings satisfying the following conditions has the integer decomposition property: 1. the size is ℓ; and By taking V ′ = ∅ in Theorem 8, we obtain the following, which is an extension of the result of McDiarmid [25].
++ , and ℓ ∈ Z + . Then, the convex hull of the incidence vectors of the b-branchings of size ℓ has the integer decomposition property.
We can also remove Condition 1 in Theorem 8.

Then, the convex hull of the incidence vectors of the b-branchings for which the indegree of each vertex
has the integer decomposition property.

Organization of the Paper
The remainder of the paper is organized as follows. Section 2 is devoted to a proof for Theorem 5 on equitable partition into matching forests. In Section 3, we prove Theorem 6 on equitable partition into b-branchings, and then derive Theorems 8-10 on the integer decomposition property of the related polytopes. This fact is extended to the following lemma on the partition of the arc set into two branchings, which can be derived from Edmonds' disjoint branchings theorem [13].

Equitable Partition into Matching Forests
Lemma 11 (Schrijver [28]). Let D = (V, A) be a digraph, and B 1 and B 2 are branchings in D partitioning A. Then, for two vertex sets , the arc set A can be partitioned into two branchings B ′ 1 and B ′ 2 such that We remark that Schrijver [29] derived Theorem 1 from Lemma 11. Here we prove that Lemma 11 further leads to Theorem 5.
Proof of Theorem 5: The case k = 1 is trivial, and thus let k ≥ 2. Let F 1 , . . . , F k be matching forests minimizing among those partitioning E ∪ A. We prove that every pair of F i and F j (i, j ∈ [k]) attains (3). Suppose to the contrary that (3) does not hold for some i, j ∈ [k]. Without loss of generality, assume Let , then v has no incoming arc in A ′ , and hence v itself forms a source component in (V, A ′ ). Thus, for X ∈ X ′ with |X| ≥ 2, it follows that X ∩ R(B 1 ) and X ∩ R(B 2 ) are not empty and disjoint with each other. Denote the family of such X ∈ X ′ by X ′′ , i.e., X ′′ = {X ∈ X ′ | |X| ≥ 2}. For each X ∈ X ′′ , take a pair e X of vertices of which one vertex is in R(B 1 ) and the other in R(B 2 ). Denote N = {e X | X ∈ X ′′ }. Note that N is a matching. Consider an undirected graph H = (V, M 1 ∪ M 2 ∪ N ). Observe that each vertex v ∈ V has degree at most two: if a vertex v ∈ V is covered by both M 1 and M 2 , then it follows that v ∈ R(B 1 ) ∩ R(B 2 ), implying that v is not covered by N . Thus, H consists of a disjoint collection of paths, some of which are possibly isolated vertices, and cycles.

Proposition 1. For a vertex v in H with degree exactly two, it holds that
Proof of Proposition 1: It is clear if v ∈ ∂M 1 ∩ ∂M 2 , and suppose not. Without loss of generality, assume v ∈ ∂M 1 ∩ ∂N . It then follows from v ∈ ∂M 1 that v ∈ R(B 1 ). Since v ∈ ∂N , this implies that v ∈ R(B 2 ). We thus conclude v ∈ ∂B 2 ⊆ ∂F 2 .
By Proposition 1, each vertex u ∈ ∂F 1 △∂F 2 is an endpoint of a path in H. It then follows from (7) that there must exist a path P such that • one endpoint u of P belongs to ∂F 1 \ ∂F 2 , and • the other endpoint u ′ of P belongs to ∂F 1 .
We remark that u ′ may or may not belong to ∂F 2 . It may also be the case that u ′ is null, i.e., u is an isolated vertex which by itself forms P .
Denote the set of vertices in P by V (P ), and the set of edges in P belonging to M 1 ∪ M 2 by E(P ). Define M ′ 1 = M 1 △E(P ) and M ′ 2 = M 2 △E(P ). It follows that M ′ 1 and M ′ 2 are matchings satisfying Also define

It then follows that
. It also follows from the construction of H that R ′ i ∩ X = ∅ for every X ∈ X ′ and for i = 1, 2. Thus, by Lemma 11, the arc set A ′ can be partitioned into two branchings B ′ 1 and B ′ 2 such that R(B ′ 1 ) = R ′ 1 and R(B ′ 2 ) = R ′ 2 . Then it holds that Moreover, by the definition of P , we have that and in particular, by (7), It also follows that This contradicts the fact that F 1 , . . . , F k minimize (6), and thus completes the proof of the theorem.
We remark that, if an arbitrary partition of E ∪ A into k matching forests is given, a partition of E ∪ A into k matching forests satisfying (3) can be found in polynomial time. This can be done by repeatedly applying the update of two matching forests described in the above proof. The time complexity follows from the fact that each update can be done in polynomial time and decreases the value (6) by at least two.

Equitable Partition into b-branchings
In this section we first prove Theorem 6, and then derive Theorems 8-10. In proving Theorem 6, we make use of the following lemma, which is an extension of Lemma 11 to b-branchings.
We now prove Theorem 6.
Proof of Theorem 6: The case k = 1 is trivial, and thus let k ≥ 2. Let B 1 , . . . , B k be k b-branchings minimizing or there exist j ∈ [k] and v ∈ V such that Without loss of generality, let i = 1 and j = 2, and denote b 1 = d − B1 and b 2 = d − B2 . Let D ′ = (V, B 1 ∪ B 2 ). Since B 1 and B 2 are b-branchings, it directly follows the definition of b-branchings that Let X be the set of source components X in D ′ such that b 1 (X) + b 2 (X) is even, and let Y be the set of source components Y in Now it directly follows from (11) It then follows from Lemma 12 that there exist b-branchings B ′ 1 and For these two b-branchings B ′ 1 and B ′ 2 , we have that Therefore, we can strictly decrease the value (8) by replacing B 1 and B 2 , which satisfy (9) or (10), with B ′ 1 and B ′ 2 , which satisfy (15). This contradicts the fact that B 1 , . . . , B k minimize (8). We remark that a partition of A into k b-branchings satisfying Conditions 1 and 2 in Theorem 6 can be found in polynomial time. First, we can check if there exists a partition of A into k b-branchings and find one if exists in polynomial time [22]. If this partition does not satisfy Conditions 1 and 2, then we repeatedly apply the update of two b-branchings as shown in the above proof, which can be done in polynomial time and strictly decreases the value (8).
We conclude this paper by deriving Theorems 8-10 from Theorem 6. Here we only present a proof for Theorem 8: Theorem 9 is a special case X = ∅ of Theorem 8; and Theorem 10 can be proved by the same argument.
Proof of Theorem 8: Denote the convex hull of the b-branchings in D by P , and that of the b-branchings in D satisfying 1 and 2 by Q. Take a positive integer κ ∈ Z ++ , and let x be an integer vector in κQ. For a vertex v ∈ V , let δ − (v) ⊆ A denote the set of arcs whose head is v. Observe that follow from the definition of Q.
Let D ′ = (V, A ′ ) be a digraph obtained from D by replacing each arc a ∈ A by x a parallel arcs. Then, since x ∈ κQ ⊆ κP , it follows from the integer decomposition property of the b-branching polytope (Theorem 7) that x is the sum of the incidence vectors of κ b-branchings, i.e., A ′ can be partitioned into κ b-branchings. Here it directly follows from (16) and (17) that and thus it follows from Theorem 6 that A ′ can be partitioned into κ b-branchings B 1 , . . . , B κ such that Therefore, x can be represented as the sum of the incidence vectors of κ b-branchings satisfying 1 and 2, i.e., integer vectors in Q, which completes the proof.