Multidecompositions of the Balanced Complete Bipartite Graph into Paths and Stars

Abstract

Let 𝑃_𝑘 and 𝑆_𝑘 denote a path and a star with 𝑘 edges, respectively. For graphs 𝐹, 𝐺, and 𝐻, a (𝐺,𝐻)-multidecomposition of 𝐹 is a partition of the edge set of 𝐹 into copies of 𝐺 and copies of 𝐻 with at least one copy of 𝐺 and at least one copy of 𝐻. In this paper, necessary and sufficient conditions for the existence of the (𝑃_𝑘, 𝑆_𝑘)-multidecomposition of the balanced complete bipartite graph are given.

1. Introduction

Let 𝐹, 𝐺, and 𝐻 be graphs. A 𝐺-decomposition of 𝐹 is a partition of the edge set of 𝐹 into copies of 𝐺. If 𝐹 has a 𝐺-decomposition, we say that 𝐹 is 𝐺-decomposable and write 𝐺∣𝐹. A (𝐺,𝐻)-multidecomposition of 𝐹 is a partition of the edge set of 𝐹 into copies of 𝐺 and copies of 𝐻 with at least one copy of 𝐺 and at least one copy of 𝐻. If 𝐹 has a (𝐺,𝐻)-multidecomposition, we say that 𝐹 is (𝐺,𝐻)-multidecomposable and write (𝐺,𝐻)∣𝐹.

For positive integers 𝑚 and 𝑛, 𝐾_𝑚,𝑛 denotes the complete bipartite graph with parts of sizes 𝑚 and 𝑛. A complete bipartite graph is balanced if 𝑚=𝑛. A 𝑘-path, denoted by 𝑃_𝑘, is a path with 𝑘 edges. A 𝑘-star, denoted by 𝑆_𝑘, is the complete bipartite graph 𝐾_1,𝑘. A 𝑘-cycle, denoted by 𝐶_𝑘, is a cycle of length 𝑘.

𝑃_𝑘-decompositions of graphs have been a popular topic of research in graph theory. Articles of interest include [1–11]. The reader can refer to [12] for an excellent survey of this topic. Decompositions of graphs into 𝑘-stars have also attracted a fair share of interest. Articles of interest include [13–18]. The study of the (𝐺,𝐻)-multidecomposition was introduced by Abueida and Daven in [19]. Abueida and Daven [20] investigated the problem of the (𝐾_𝑘,𝑆_𝑘)-multidecomposition of the complete graph 𝐾_𝑛. Abueida and Daven [21] investigated the problem of the (𝐶₄,𝐸₂)-multidecomposition of several graph products where 𝐸₂ denotes two vertex disjoint edges. Abueida and O'Neil [22] settled the existence problem of the (𝐶_𝑘,𝑆_𝑘−1)-multidecomposition of the complete multigraph 𝜆𝐾_𝑛 for 𝑘=3,4, and 5. In [23], Priyadharsini and Muthusamy gave necessary and sufficient conditions for the existence of the (𝐺_𝑛,𝐻_𝑛)-multidecomposition of 𝜆𝐾_𝑛 where 𝐺_𝑛,𝐻_𝑛∈{𝐶_𝑛,𝑃_𝑛−1,𝑆_𝑛−1}. Furthermore, Shyu [24] investigated the problem of decomposing 𝐾_𝑛 into 𝑘-paths and 𝑘-stars, and gave a necessary and sufficient condition for 𝑘=3. In [25], Shyu considered the existence of a decomposition of 𝐾_𝑛 into 𝑘-paths and 𝑘-cycles and established a necessary and sufficient condition for 𝑘=4. He also gave criteria for the existence of a decomposition of 𝐾_𝑛 into 3-paths and 3 cycles in [26]. Shyu [27] investigated the problem of decomposing 𝐾_𝑛 into 𝑘-cycles and 𝑘-stars and settled the case 𝑘=4. Recently, Lee [28] established necessary and sufficient conditions for the existence of the (𝐶_𝑘,𝑆_𝑘)-multidecomposition of a complete bipartite graph.

In this paper, we investigate the problem of the (𝑃_𝑘,𝑆_𝑘)-multidecomposition of the balanced complete bipartite graph and give necessary and sufficient conditions for such a multidecomposition to exist.

2. Preliminaries

For our discussions, some terminologies and notations are needed. Let 𝐺 be a graph. The degree of a vertex 𝑥 of 𝐺, denoted by deg_𝐺𝑥, is the number of edges incident with 𝑥. A graph is 𝑟-regular if each vertex is of degree 𝑟. The vertex of degree 𝑘 in 𝑆_𝑘 is called the center of 𝑆_𝑘. Let 𝑉 and 𝐸 be subsets of the vertex set and the edge set of 𝐺, respectively. We use 𝐺[𝑉] to denote the subgraph of 𝐺 induced by 𝑉 and 𝐺−𝐸 to denote the subgraph obtained from 𝐺 by deleting 𝐸. Suppose that 𝐺₁,𝐺₂,…,𝐺_𝑡 are edge-disjoint-graphs. Then, 𝐺₁+𝐺₂+⋯+𝐺_𝑡, or ∑^𝑡_𝑖=1𝐺_𝑖, denotes the graph 𝐻 with vertex set 𝑉(𝐻)=⋃^𝑡_𝑖=1𝑉(𝐺_𝑖), and edge set 𝐸(𝐻)=⋃^𝑡_𝑖=1𝐸(𝐺_𝑖). Thus, if a graph 𝐻 can be decomposed into subgraphs 𝐺₁,𝐺₂,…,𝐺_𝑡, we write 𝐻=𝐺₁+𝐺₂+⋯+𝐺_𝑡, or 𝐻=∑^𝑡_𝑖=1𝐺_𝑖. Moreover, ⌈𝑥⌉ denotes the smallest integer not less than 𝑥 and ⌊𝑥⌋ denotes the largest integer not greater than 𝑥. Let 𝑣₀𝑣₁𝑣₂,…,𝑣_𝑘 denote the 𝑘-path with edges 𝑣₀𝑣₁,𝑣₁𝑣₂,…,𝑣_𝑘−1𝑣_𝑘, and (𝑣₁,𝑣₂,…,𝑣_𝑘) denote the 𝑘-cycle with edges 𝑣₁𝑣₂,𝑣₂𝑣₃,…,𝑣_𝑘−1𝑣_𝑘,𝑣_𝑘𝑣₁. Throughout the paper, (𝐴,𝐵) denotes the bipartition of 𝐾_𝑛,𝑛, where 𝐴={𝑎₀,𝑎₁,…,𝑎_𝑛−1} and 𝐵={𝑏₀,𝑏₁,…,𝑏_𝑛−1}.

For the edge 𝑎_𝑖𝑏_𝑗 in 𝐾_𝑛,𝑛, the label of 𝑎_𝑖𝑏_𝑗 is 𝑗−𝑖(mod𝑛). For example, in 𝐾_9,9 the labels of 𝑎₃𝑏₇ and 𝑎₈𝑏₂ are 4 and 3, respectively. Note that each vertex of 𝐾_𝑛,𝑛 is incident with exactly one edge with label 𝑖 for 𝑖=0,1,2,…,𝑛−1. Let 𝐺 be a subgraph of 𝐾_𝑛,𝑛 and 𝑡 a nonnegative integer. We use 𝐺_+𝑡 to denote the graph with vertex set {𝑎_𝑖∶𝑎_𝑖∈𝑉(𝐺)}⋃{𝑏_𝑗+𝑡∶𝑏_𝑗∈𝑉(𝐺)} and edge set {𝑎_𝑖𝑏_𝑗+𝑡∶𝑎_𝑖𝑏_𝑗∈𝐸(𝐺)}, where the subscripts of 𝑏 are taken modulo 𝑛. In particular, 𝐺₊₀=𝐺.

The following results due to Yamamoto et al. and Parker, respectively, are essential for our discussions.

Proposition 1 (see [18]). Let 𝑚≥𝑛≥1 be integers. Then, 𝐾_𝑚,𝑛 is 𝑆_𝑘-decomposable if and only if 𝑚≥𝑘 and ()()𝑚≡0mod𝑘𝑖𝑓𝑛<𝑘,𝑚𝑛≡0mod𝑘𝑖𝑓𝑛≥𝑘. (1)

Proposition 2 (see [7]). There exists a 𝑃_𝑘-decomposition of 𝐾_𝑚,𝑛 if and only if 𝑚𝑛≡0(mod𝑘), and one of aforementioned (see Table 1) cases occurs.

Table 1 

3. Main Results

We first give necessary conditions of the (𝑃_𝑘,𝑆_𝑘)-multidecomposition of 𝐾_𝑛,𝑛.

Lemma 3. Let 𝑘 and 𝑛 be positive integers. If there exists a (𝑃_𝑘,𝑆_𝑘)-multidecomposition of 𝐾_𝑛,𝑛, then 𝑘≤𝑛 and 𝑛²≡0(mod𝑘).

Proof. The result follows from the fact that the maximum size of a star in 𝐾_𝑛,𝑛 is 𝑛, the size of each member in the multidecomposition is 𝑘, and |𝐸(𝐾_𝑛,𝑛)|=𝑛².

We now show that the necessary conditions are also sufficient. Since 𝑃_𝑘=𝑆_𝑘 for 𝑘=1,2, the result holds for 𝑘=1,2 by Proposition 1. So it remains to consider the case 𝑘≥3. The proof is divided into cases 𝑛≥2𝑘, 𝑛=𝑘 and 𝑘<𝑛<2𝑘, which are treated in Lemmas 4, 5, and 6, respectively.

Lemma 4. Let 𝑘 and 𝑛 be positive integers with 𝑛≥2𝑘≥6. If 𝑛²≡0(mod𝑘), then 𝐾_𝑛,𝑛 is (𝑃_𝑘,𝑆_𝑘)-multidecomposable.

Proof. Let 𝑛=𝑞𝑘+𝑟 where 𝑞 and 𝑟 are integers with 0≤𝑟<𝑘. Then, 𝑞≥2 from the assumption 𝑛≥2𝑘. Note that 𝐾_𝑛,𝑛=𝐾_{𝑞𝑘+𝑟,𝑛}=𝐾_{(𝑞−1)𝑘,𝑛}+𝐾_{𝑘+𝑟,𝑛}. (2) By Proposition 1, 𝐾_{(𝑞−1)𝑘,𝑛} is 𝑆_𝑘-decomposable. On the other hand, trivially, 𝑘≤min{𝑘+𝑟,𝑛}, and |𝐸(𝐾_{𝑘+𝑟,𝑛})|=(𝑘+𝑟)𝑛≡0(mod𝑘) from the assumption 𝑛²≡0(mod𝑘). This implies that 𝐾_{𝑘+𝑟,𝑛} is 𝑃_𝑘-decomposable by Proposition 2. Hence, 𝐾_𝑛,𝑛 is (𝑃_𝑘,𝑆_𝑘)-multidecomposable.

Lemma 5. Let 𝑘 be a positive integer with 𝑘≥3. Then, 𝐾_𝑘,𝑘 is (𝑃_𝑘,𝑆_𝑘)-multidecomposable.

Proof. Note that 𝐾_𝑘,𝑘=𝐾_1,𝑘+𝐾_{𝑘−1,𝑘}. Trivially, 𝐾_1,𝑘=𝑆_𝑘. On the other hand, |𝐸(𝐾_{𝑘−1,𝑘})|=(𝑘−1)𝑘≡0(mod𝑘). Furthermore, 𝑘≤2(𝑘−1)−1 for odd 𝑘, and 𝑘≤2(𝑘−1)=2𝑘−2 for even 𝑘. Hence, 𝐾_{𝑘−1,𝑘} is 𝑃_𝑘-decomposable by Proposition 2, and 𝐾_𝑘,𝑘 is (𝑃_𝑘,𝑆_𝑘)-multidecomposable.

Lemma 6. Let 𝑘 and 𝑛 be integers with 3≤𝑘<𝑛<2𝑘. If 𝑛²≡0(mod𝑘), then 𝐾_𝑛,𝑛 is (𝑃_𝑘,𝑆_𝑘)-multidecomposable.

Proof. Suppose that 𝑛=𝑘+𝑟. Then, 0<𝑟<𝑘 from the assumption 𝑘<𝑛<2𝑘. Let 𝐴₁={𝑎₀,𝑎₁,…,𝑎_𝑘−1}, 𝐵₁={𝑏₀,𝑏₁,…,𝑏_𝑘−1}, 𝐴₂={𝑎_𝑘,𝑎_𝑘+1,…,𝑎_{𝑘+𝑟−1}}, and 𝐵₂={𝑏_𝑘,𝑏_𝑘+1,…,𝑏_{𝑘+𝑟−1}}. Let 𝐺_𝑖=𝐾_𝑛,𝑛[𝐴_𝑖∪𝐵₁] for 𝑖=1,2 and 𝐻=𝐾_𝑛,𝑛[𝐴∪𝐵₂]. Then, 𝐾_𝑛,𝑛=𝐺₁+𝐺₂+𝐻. Note that 𝐺₁ is isomorphic to 𝐾_𝑘,𝑘, 𝐻 is isomorphic to 𝐾_{𝑘+𝑟,𝑟}, and 𝐺₂ is isomorphic to 𝐾_𝑟,𝑘, which is 𝑆_𝑘-decomposable by Proposition 1. Hence, it is sufficient to show that 𝐺₁+𝐻 is (𝑃_𝑘,𝑆_𝑘)-multidecomposable.
Let 𝑡=𝑟²/𝑘. Since 𝑘∣𝑛², we have 𝑘∣𝑟², which implies that 𝑡 is a positive integer Let 𝐶=(𝑏₀,𝑎₀,𝑏₁,𝑎₁,…,𝑏_𝑘−2,𝑎_𝑘−2,𝑏_𝑘−1,𝑎_𝑘−1). Then, 𝐶 is a 2𝑘-cycle in 𝐺₁. Let 𝑝=⌊𝑡/2⌋. For odd 𝑡, define a 𝑘-path 𝑃 in 𝐺₁ as follows: 𝑏𝑃=^2𝑝𝑎₀𝑏_2𝑝+1𝑎₁⋯𝑏_{2𝑝−1+𝑘/2}𝑎_𝑘/2−1𝑏_2𝑝+𝑘/2𝑏if𝑘iseven,_2𝑝𝑎₀𝑏_2𝑝+1𝑎₁⋯𝑏_{2𝑝+(𝑘−1)/2}𝑎_(𝑘−1)/2if𝑘isodd, (3) where the subscripts of 𝑏 are taken modulo 𝑘. Since 𝑡=𝑟²/𝑘 and 𝑟<𝑘, we have 𝑡≤𝑟−1≤𝑘−2. (4) Thus, 2𝑝+1=𝑡≤𝑘−2, which implies the labels of the edges in 𝑃 are 2𝑝 and 2𝑝+1. Note that for 𝑖=0,1,…,𝑝−1, 𝐶_+2𝑖 is a 2𝑘-cycle which consists of all of the edges with labels 2𝑖 and 2𝑖+1 in 𝐺₁. Thus, 𝐶,𝐶₊₂,…,𝐶_+2(𝑝−1) and 𝑃 are edge-disjoint in 𝐺₁.
Define a subgraph 𝑊 of 𝐺₁ as follows: 𝑊=^𝑝−1_𝑖∑₌₀𝐶_+2𝑖foreven𝑡,_𝑝−1𝑖∑₌₀𝐶_+2𝑖+𝑃forodd𝑡≥3,𝑃for𝑡=1. (5) Since 𝐶_2𝑘 can be decomposed into 2 copies of 𝑃_𝑘 and 2𝑝=𝑡 for even 𝑡 as well as 2𝑝+1=𝑡 for odd 𝑡, 𝑊 can be decomposed into 𝑡 copies of 𝑃_𝑘. Let 𝛿=1 for even 𝑘 and 𝛿=0 for odd 𝑘. Note that for even 𝑡, deg_{𝐺1−𝐸(𝑊)}𝑎_𝑖=𝑘−2𝑝=𝑘−𝑡, and for odd 𝑡, deg_{𝐺1−𝐸(𝑊)}𝑎_𝑖=𝑘𝑘−𝑡−1if𝑖=0,1,…,2𝑘−2,𝑘−𝑡−𝛿if𝑖=2𝑘−1,𝑘−𝑡+1if𝑖=2,𝑘2+1,…,𝑘−1. (6) Let 𝑋_𝑖=(𝐺₁−𝐸(𝑊))[{𝑎_𝑖}∪𝐵₁] for 𝑖=0,1,…,𝑘−1. Then for even 𝑡, 𝑋_𝑖=𝑆_𝑘−𝑡, and for odd 𝑡, 𝑋_𝑖=𝑆_{𝑘−𝑡−1}𝑘if𝑖=0,1,…,2𝑆−2,_{𝑘−𝑡−𝛿}𝑘if𝑖=2𝑆−1,_{𝑘−𝑡+1}𝑘if𝑖=2,𝑘2+1,…,𝑘−1 (7) with the center at 𝑎_𝑖. In the following, we will show that 𝐻 can be decomposed into 𝑟 copies of 𝑆_𝑘 with centers in 𝐵₂, and into 𝑘 copies of 𝑆_𝑡 with centers in 𝐴₁ for even 𝑡, and into ⌈𝑘/2⌉−1 copies of 𝑆_𝑡+1 with centers in {𝑎₀,𝑎₁,…,𝑎_{⌈𝑘/2⌉−2}}, an 𝑆_𝑡+𝛿 with the center at 𝑎_{⌈𝑘/2⌉−1}, and 𝑘−⌈𝑘/2⌉ copies of 𝑆_𝑡−1 with centers in {𝑎_⌈𝑘/2⌉,𝑎_{⌈𝑘/2⌉+1},…,𝑎_𝑘−1} for odd 𝑡.
We show the required star decomposition of 𝐻 by orienting the edges of 𝐻. For any vertex 𝑥 of 𝐻, the outdegree deg⁺_𝐻𝑥 (indegree deg⁻_𝐻𝑥, resp.) of 𝑥 in an orientation of 𝐻 is the number of arcs incident from (to, resp.) 𝑥. It is sufficient to show that there exists an orientation of 𝐻 such that deg⁺_𝐻𝑏_𝑗=𝑘, (8) where 𝑗=𝑘,𝑘+1,…,𝑘+𝑟−1, and for even 𝑡deg⁺_𝐻𝑎_𝑖=𝑡, (9) where 𝑖=0,1,…,𝑘−1, and for odd 𝑡deg⁺_𝐻𝑎_𝑖=𝑘𝑡+1if𝑖=0,1,…,2𝑘−2,𝑡+𝛿if𝑖=2𝑘−1,𝑡−1if𝑖=2,𝑘2+1,…,𝑘−1. (10)
We first consider the edges oriented outward from 𝐴₁ according to the parity of 𝑡. Let 𝛽=𝑘+(𝑡+1)(⌈𝑘/2⌉−1) and 𝛾=𝛽+𝑡+𝛿. If 𝑡 is even, then the edges 𝑎_𝑖𝑏_{𝑘+𝑡𝑖},𝑎_𝑖𝑏_{𝑘+𝑡𝑖+1},…,𝑎_𝑖𝑏_{𝑘+𝑡𝑖+𝑡−1} are all oriented outward from 𝑎_𝑖, where 𝑖=0,1,…,𝑘−1. If 𝑡 is odd, then the edges 𝑎_𝑖𝑏_{𝑘+(𝑡+1)𝑖},𝑎_𝑖𝑏_{𝑘+(𝑡+1)𝑖+1},…,𝑎_𝑖𝑏_{𝑘+(𝑡+1)𝑖+𝑡} for 𝑖=0,1,…,⌈𝑘/2⌉−2, and 𝑎_{⌈𝑘/2⌉−1}𝑏_𝛽,𝑎_{⌈𝑘/2⌉−1}𝑏_𝛽+1,…, 𝑎_{⌈𝑘/2⌉−1}𝑏_{𝛽+𝑡+𝛿−1}, as well as 𝑎_𝑖𝑏_{(𝑡−1)(𝑖−⌈𝑘/2⌉)+𝛾},𝑎_𝑖𝑏_{(𝑡−1)(𝑖−⌈𝑘/2⌉)+𝛾+1},…,𝑎_𝑖𝑏_{(𝑡−1)(𝑖−⌈𝑘/2⌉)+𝛾+𝑡−2} for 𝑖=⌈𝑘/2⌉,⌈𝑘/2⌉+1,…,𝑘−1 are all oriented outward from 𝑎_𝑖. In both cases, the subscripts of 𝑏 are taken modulo 𝑟 in the set of numbers {𝑘,𝑘+1,…,𝑘+𝑟−1}. Note that for even 𝑡 we orient 𝑡 edges from each 𝑎_𝑖 and for odd 𝑡 we orient at most 𝑡+1 edges from 𝑎_𝑖. By inequality (4), we have 𝑡+1≤𝑟, which assures us that there are enough edges for the above orientation.
Finally, the edges which are not oriented yet are all oriented from 𝐵₂ to 𝐴. From the construction of the orientation, it is easy to see that (9) and (10) are satisfied, and for all 𝑏_𝑤,𝑏_𝑤′∈𝐵₂, we have deg⁻_𝐻𝑏_𝑤−deg⁻_𝐻𝑏_𝑤′≤1. (11) So, we only need to check (8).
Since deg⁺_𝐻𝑏_𝑤+deg⁻_𝐻𝑏_𝑤=𝑘+𝑟 for 𝑏_𝑤∈𝐵₂, it follows from (11) that deg⁺_𝐻𝑏_𝑤−deg⁺_𝐻𝑏_𝑤′≤1 (12) for 𝑏_𝑤,𝑏_𝑤′∈𝐵₂. Note that for even 𝑡, ∑^𝑘−1_𝑖=0deg⁺_𝐻𝑎_𝑖=𝑡𝑘, and for odd 𝑡, ^𝑘−1_𝑖=0deg⁺_𝐻𝑎_𝑖=()𝑘𝑡+12()𝑘−1+𝑡+𝛿+𝑡−1𝑘−2=()𝑘𝑡−1𝑘+22=()𝑘−1+𝛿𝑡−1𝑘+22()2()−1+1if𝑘iseven,𝑡−1𝑘+𝑘+12−1if𝑘isodd.=𝑡𝑘. (13) Thus, ^{𝑘+𝑟−1}_𝑤=𝑘deg⁺_𝐻𝑏_𝑤=𝐸𝐾_{𝑘+𝑟,𝑟}−^𝑘−1_𝑖=0deg⁺_𝐻𝑎_𝑖=(𝑘+𝑟)𝑟−𝑡𝑘=𝑘𝑟+𝑟²−𝑟²=𝑘𝑟. (14) Therefore from (12), we have deg⁺_𝐻𝑏_𝑤=𝑘 for 𝑏_𝑤∈𝐵₂. This establishes (8). Hence, there exists the required decomposition 𝒟of 𝐻. Let 𝑋^′_𝑖 be the star with center at 𝑎_𝑖 in 𝒟 for 𝑖=0,1,…,𝑘−1. Then, 𝑋_𝑖+𝑋^′_𝑖 is a 𝑘-star. This completes the proof.

Now, we are ready for the main result. It is obtained form the arguments above, Lemma 4 and Lemmas 3, 4, 5, and 6.

Theorem 7. Let 𝑘 and 𝑛 be positive integers. Then, 𝐾_𝑛,𝑛 has a (𝑃_𝑘,𝑆_𝑘)-multidecomposition if and only if 𝑘≤𝑛 and 𝑛²≡0(mod𝑘).

Acknowledgment

The authors are grateful to the referees for the valuable comments.