On the Strong Equitable Vertex 2-Arboricity of Complete Bipartite Graphs

: An equitable partition of a graph G is a partition of the vertex set of G such that the sizes of any two parts differ by at most one. The strong equitable vertex k - arboricity of G , denoted by va ≡ k ( G ) , is the smallest integer t such that G can be equitably partitioned into t (cid:48) induced forests for every t (cid:48) ≥ t , where the maximum degree of each induced forest is at most k . In this paper, we provide a general upper bound for va ≡ 2 ( K n , n ) . Exact values are obtained in some special cases.


Introduction
All graphs considered in this paper are finite, undirected and simple. For a real number x, x is the least integer not less than x and x is the largest integer not larger than x. For a graph G, we use V(G) and E(G) to denote the vertex set and edge set, respectively. For a complete bipartite graph K n,n , we let X = {x 1 , . . . , x n } and Y = {y 1 , . . . , y n } be the partite sets of K n,n . For other undefined concepts, we refer the readers to [1].
The vertex arboricity of a graph G, denoted by va(G), is the minimum number of subsets into which the vertex set V(G) can be partitioned so that each subset induces a forest. This notion was first introduced by Chartrand and Kronk in 1969 [2].
Wu, Zhang and Li [3] introduced an equitable version of the vertex arboricity in 2013. An equitable partition of a graph G is a partition of the vertex set of G such that the sizes of any two parts differ by at most one. The equitable partition problem was first introduced by Meyer [4], motivated by a practical application to municipal garbage collection [5]. In this context, the vertices of the graph represent garbage collection routes. A pair of vertices share an edge if the corresponding routes should not be run on the same day. It is desirable that the number of routes run on each day be approximately the same. Therefore, the problem of assigning one of the six weekly working days to each route reduces to finding a proper equitable partition which has six parts. The equitable partition problem has many applications such as in scheduling, constructing timetables and load balance in parallel memory systems, please see [6][7][8][9][10][11].
Moreover, Wu, Zhangf and Li [3] consider a restriction on the maximum degree of each induced forest. The equitable vertex k-arboricity of a graph G, denoted by va = k (G), is the minimum number of induced forests into which G can be equitably partitioned, where the maximum degree of each induced forest is at most k. The strong equitable vertex k-arboricity of G, denoted by va ≡ k (G), is the smallest integer t such that G can be equitably partitioned into t induced forests for every t ≥ t, where the maximum degree of each induced forest is at most k.
Note that va = k (G) and va ≡ k (G) may vary greatly. For example, it is easy to see that va = ∞ (K n,n ) = 2, but va ≡ ∞ (K n,n ) = 2 ( √ 8n + 9 − 1)/4 if 2n = t(t + 3) and t is odd (see [3]). Wu et al. [3] first investigated the strong equitable vertex 1-arboricity of complete bipartite graphs. They provided a sharp upper bound for va ≡ 1 (K n,n ) in general case: Noting that the upper bound given in Lemma 1 is not very tight for some special graphs, Wu et al. [3] commented that determining the strong equitable 1-arboricity for every K n,n seems not to be an easy task. Furthermore, the exact values of some special cases were studied by Wu et al. [3] and Tao [12]. Concerning va ≡ ∞ (G), Wu et al. [3] posed the following conjectures: for every graph.

Conjecture 2. ([3])
There is a constant c such that va ≡ ∞ (G) ≤ c for every planar graph G.
In this paper, we focus on the strong equitable vertex 2-arboricity of K n,n . In Section 2, we provide a general upper bound for va ≡ 2 (K n,n ). In Sections 3-5, we make our efforts to improve this upper bound where n ≡ 0, 1, 2(mod3). In some special cases, the exact values of va ≡ 2 (K n,n ) are determined.

Upper
Bound for va ≡ 2 (K n,n ) In this section, we provide an upper bound for va ≡ 2 (K n,n ).

Lemma 2. ([3])
The complete bipartite graph K n,n can be equitably partitioned into t induced forests with the maximum degree at most k for every even integer t ≥ 2. Theorem 1. Let K n,n be a complete bipartite graph. Then, Proof. It suffices to show that K n,n can be equitably partitioned into q induced forests for every q ≥ 2 n+1 3 , where the maximum degree of each forest is at most 2. Let n+1 3 = m. We split the proof into three parts: We partition V(K n,n ) into q subsets equitably, that is, each subset has size 2n/q or 2n/q . Since q ≥ 2m, then 2n q ≤ 2n 2m = 6m−2 2m < 3, hence each subset contains no more than three vertices. It follows that each subset induces a forest with the maximum degree at most 2.
Case 2. n + 1 = 3m + 1. We partition V(K n,n ) into q subsets equitably. Since 2n q ≤ 2n 2m = 6m 2m = 3, one can easily check that each subset induces a forest with the maximum degree at most 2.
It is worth noting that the upper bound for va ≡ 2 (K n,n ) is the same as that for va ≡ 1 (K n,n ). The upper bound in Theorem 1 is helpful to determine the exact value of va ≡ 2 (K n,n ) for small n. For example, we can deduce from Theorem 1 that va ≡ 2 (K 6,6 ) = 4(It is not difficult to check that va ≡ 2 (K 6,6 ) > 3). However, for large n, we need to improve this bound.
The following lemma is useful in our proofs.

Lemma 3.
If the cardinality of a subset is at least 4 in any equitable partition of K n,n where each subset induces a forest with the maximum degree at most 2, then the vertices of this subset belong to the same partite set.
Proof. Suppose, to the contrary, that all the vertices of this subset belong to different partite sets of K n,n . Since it contains at least four vertices, the subgraph induced by all the vertices of this subset either has a cycle or has maximum degree at least 3, a contradiction.

va
Proof. We prove it by induction on m. If m = 0, the result holds by Theorem 1. Suppose m ≥ 1.
Since t ≥ 4m > 4(m − 1), by Lemma 2 and the induction hypothesis, it suffices to show that K 3t,3t can be equitably partitioned into 2t − 2m + 1 induced forests with the maximum degree of each forest at most 2. We equitably partition X into t − m subsets and Y into t − m + 1 subsets, respectively. It is straightforward to check that each subset contains either three or four vertices and each subset forms an independent set. The proof of the theorem is complete.
Compared with Theorem 1, Theorem 2 improves the upper bound for va ≡ 2 (K n,n ) greatly, especially for large n. For example, we can deduce from Theorem 1 that va ≡ 2 (K 48,48 ) ≤ 2 × 16 = 32. In fact, this upper bound can be improved to 24 using Theorem 2.
In the following, we make our efforts to determine the exact values of va ≡ 2 (K 3t,3t ) in some special cases.

va
Proof. We prove this theorem by induction on m. If m = 0, the result holds by Theorem 1. Suppose m ≥ 1. Since t ≥ 4m + 1 > 4(m − 1) + 1, by Lemma 2 and the induction hypothesis, it suffices to show that K 3t+1,3t+1 can be equitably partitioned into 2t − 2m + 1 induced forests with the maximum degree of each forest at most 2. We equitably partition X into t − m subsets and Y into t − m + 1 subsets, respectively. It is straightforward to check that each subset contains either three or four vertices and each subset forms an independent set. The proof of the theorem is complete.
For some special cases, the exact values of va ≡ 2 (K 3t+1,3t+1 ) can be determined by Theorem 5 and Lemma 3.

va
We first prove a useful lemma.
Proof. The proof is similar to that of Theorems 2 and 5, which is omitted here.