Packing and covering the balanced complete bipartite multigraph with cycles and stars

Let C k denote a cycle of length k and let S k denote a star with k edges. For multigraphs F , G and H , an ( F, G ) - decomposition of H is an edge decomposition of H into copies of F and G using at least one of each. For L ⊆ H and R ⊆ rH , an ( F, G ) -packing (resp. ( F, G ) -covering) of H with leave L (resp. padding R ) is an ( F, G ) - decomposition of H − E ( L ) (resp. H+E(R)). An ( F, G ) -packing (resp. ( F, G ) -covering) of H with the largest (resp. smallest) cardinality is a maximum ( F, G ) -packing (resp. minimum ( F, G ) -covering), and its cardinality is referred to as the ( F, G ) -packing number (resp. ( F, G ) -covering number) of H . In this paper, we determine the packing number and the covering number of λK n,n with C k ’s and S k ’s for any λ , n and k , and give the complete solution of the maximum packing and the minimum covering of λK n,n with 4 -cycles and 4 -stars for any λ and n with all possible leaves and paddings.


Introduction
For positive integers m and n, K m,n denotes the complete bipartite graph with parts of sizes m and n.If m = n, the complete bipartite graph is referred to as balanced.A k-cycle, denoted by C k , is a cycle of length k.A k-star, denoted by S k , is the complete bipartite graph K 1,k .A k-path, denoted by P k , is a path with k vertices.For a graph H and a positive integer λ, we use λH to denote the multigraph obtained from H by replacing each edge e by λ edges each having the same endpoints as e.When λ = 1, 1H is simply written as H.
Let F , G, and H be multigraphs.A decomposition of H is a set of edge-disjoint subgraphs of H whose union is H.An (F, G)-decomposition of H is a decomposition of H into copies of F and G using at least one of each.If H has an (F, G)-decomposition, we say that H is (F, G)-decomposable and write (F, G)|H.If H does not admit an (F, G)-decomposition, two natural questions arise: (1) What is the minimum number of edges needed to be removed from the edge set of H so that the resulting graph is (F, G)-decomposable, and what does the collection of removed edges look like?
(2) What is the minimum number of edges needed to be added to the edge set of H so that the resulting graph is (F, G)-decomposable, and what does the collection of added edges look like?
These questions are respectively called the maximum packing problem and the minimum covering problem of H with F and G.
Let F , G, and H be multigraphs.For L ⊆ H and R ⊆ rH, an (F, G)-packing of H with leave L is an (F, G)-decomposition of H − E(L), and an (F, G)-covering with padding R is an (F, G)-decomposition of H + E(R).For an (F, G)-packing P of H with leave L, if |P| is as large as possible (so that |L| is as small as possible), then P and L are referred to as a maximum (F, G)-packing and a minimum leave, respectively.Moreover, the cardinality of the maximum (F, G)-packing of H is called the (F, G)-packing number of H, denoted by p(H; F, G).For an (F, G)-covering C of H with padding R, if |C | is as small as possible (so that |R| is as small as possible), then C and R are referred to as a minimum covering and a minimum padding, respectively.Moreover, the cardinality of the minimum (F, G)-covering of H is called the (F, G)-covering number of H, denoted by c(H; F, G).Clearly, an (F, G)-decomposition of H is a maximum (F, G)-packing with leave the empty graph, and also a minimum (F, G)-covering with padding the empty graph.
Recently, decomposition into a pair of graphs has attracted a fair share of interest.Abueida and Daven [3] investigated the problem of (K k , S k )-decomposition of the complete graph K n .Abueida and Daven [4] investigated the problem of the (C 4 , E 2 )-decomposition of several graph products where E 2 denotes two vertex disjoint edges.Abueida and O'Neil [7] settled the existence problem for (C k , S k−1 )decomposition of the complete multigraph λK n for k ∈ {3, 4, 5}.Priyadharsini and Muthusamy [12,13] gave necessary and sufficient conditions for the existence of (G n , H n )-decompositions of λK n and λK n,n where G n , H n ∈ {C n , P n , S n−1 }.A graph-pair (G, H) of order m is a pair of non-isomorphic graphs G and H on m non-isolated vertices such that G ∪ H is isomorphic to K m .Abueida and Daven [2] and Abueida, Daven and Roblee [5] completely determined the values of n for which λK n admits a (G, H)decomposition where (G, H) is a graph-pair of order 4 or 5. Abueida, Clark and Leach [1] and Abueida and Hampson [6] considered the existence of decompositions of K n − F for the graph-pair of order 4 and 5, respectively, where F is a Hamiltonian cycle, a 1-factor, or almost 1-factor.Furthermore, Shyu [14] investigated the problem of decomposing K n into paths and stars with k edges, giving a necessary and sufficient condition for k = 3.In [15,16], Shyu considered the existence of a decomposition of K n into paths and cycles with k edges, giving a necessary and sufficient condition for k ∈ {3, 4}.Shyu [17] investigated the problem of decomposing K n into cycles and stars with k edges, settling the case k = 4.In [18], Shyu considered the existence of a decomposition of K m,n into paths and stars with k edges, giving a necessary and sufficient condition for k = 3.Recently, Lee [9] and Lee and Lin [10] established necessary and sufficient conditions for the existence of (C k , S k )-decompositions of the complete bipartite graph and the complete bipartite graph with a 1-factor removed, respectively.However, much less work has been done on the problem of packing and covering graphs with a pair of graphs.Abueida and Daven [3] obtained the maximum packing and the minimum covering of the complete graph K n with (K k , S k ).Abueida and Daven [2] and Abueida, Daven and Roblee [5] gave the maximum packing and the minimum covering of K n and λK n with G and H, respectively, where (G, H) is a graph-pair of order 4 or 5.In this paper, we determine the packing number and the covering number of λK n,n with k-cycles and k-stars for any λ, n and k, and give the complete solution of the maximum packing and the minimum covering of λK n,n with 4-cycles and 4-stars for any λ and n with all possible leaves and paddings.

Preliminaries
In this section we first collect some needed terminology and notation, and then present a result which is useful for our discussions to follow.
Let G be a multigraph.The degree of a vertex x of G, denoted by deg G x, is the number of edges incident with x.The vertex of degree k in S k is the center of S k and any vertex of degree 1 is an endvertex of S k .For W ⊆ V (G), we use G[W ] to denote the subgraph of G induced by W . Furthermore, µ(uv) denotes the number of edges of G joining u and v, (v 1 , . . ., v k ) and v 1 . . .v k denote the k-cycle and the k-path through vertices v 1 , . . ., v k in order, respectively, and (x; y 1 , . . ., y k ) denotes the k-star with center x and endvertices y 1 , . . ., y k .When G 1 , G 2 , . . ., G t are multigraphs, not necessarily disjoint, we write Given an S k -decomposition of G, a central function c from V (G) to the set of non-negative integers is defined as follows.For each v ∈ V (G), c(v) is the number of k-stars in the decomposition whose center is v.
The following result is essential to our proof.
where ε(S) denotes the number of edges of H with both ends in S.
In the sequel of the paper, (A, B) denotes the bipartition of λK n,n , where A = {a 0 , a 1 , . . ., a n−1 } and

Packing numbers and covering numbers
In this section the packing number and the covering number of the balanced complete bipartite multigraph with k-cycles and k-stars are determined.We begin with a criterion for decomposing the complete bipartite graph into k-cycles.Proposition 3.1 (Sotteau [19]) For positive integers m, n, and k, the graph K m,n is C k -decomposable if and only if m, n, and k are even, k ≥ 4, min{m, n} ≥ k/2, and k divides mn.
Let K * m,n denote the symmetric complete bipartite digraph with parts of size m and n, and let Proof: Since λK m,n is isomorphic to λK n,m , it suffices to show that the result holds for k | m.If λ is odd, then m and n are even from the assumption λm ≡ λn ≡ 0 (mod 2).Since k divides mn, Proposition 3.
Lemma 3.6 Let k be a positive even integer and let n be a positive integer with The assumption k < n < 2k implies 0 < r < k.We first give the required packing.Note that where ρ = 1 if s is odd, and ρ = 2 if s is even.Define a function c : V (H) → N as follows: Now we show that there exists an S k -decomposition of H with central function c by Proposition 2.1.
Define a set T of ordered pairs of vertices as follows: and for u ∈ S and For S ⊆ V (H), let µ(uv) By (1)−(3) and This settles (iii) and completes the proof. 2 Before going on, the following results are needed. Therefore Let p 0 = t/2 and p 1 = t/2 .We have p 0 = 1 and p 1 = 0 for t = 1, and In the sequel, we will show that λK n,n has a packing P consisting of t copies of k-cycles and λ(k + 2r) copies of k-stars with leave P s+1 (except in the case s = 0, in which the leave is the empty graph), and a covering Clearly, G 0 and G 1 are isomorphic to λK k/2,k .By Lemma 3.9, there exist p i edge-disjoint k-cycles in G i for i ∈ {0, 1}, and there exist p 1 + 1 edge-disjoint k-cycles in G 1 for λ ≥ 2 or r ≤ k − 2. Let δ = 0 for p 1 = 0 and δ = 1 for p 1 ≥ 1. Suppose that For s > 0, define an (s + 1)-path P as follows: where ρ = 1 if s is odd, and ρ = 2 if s is even.Define a function c : V (H) → N as follows: Now we show that there exists an S k -decomposition D of H with central function c by Proposition 2.1.
Define a set T of ordered pairs of vertices as follows: and for u ∈ S and For S ⊆ V (H), let By (4)−( 6) and In addition, ρ = 1 for odd s and ρ = 2 for even s.

Packing and covering with 4-cycles and 4-stars
In this section a complete solution to the maximum packing and minimum covering problem of λK n,n with C 4 and S 4 is given.Before that, we need more notations.For multigraphs G and H, G H denotes the disjoint union of G and H, G H denotes the union of G and H with a common vertex.For a set R and a positive integer t, tR denotes the multiset in which each element in R appears t times.In addition, M t denotes the graph induced by t nonadjacent edges.We begin with the discussion for the possible minimum leaves and paddings of λK n,n with C 4 and S 4 .
Proof: It suffices to show that K 5,5 + 3{a 0 b 0 } is not (C 4 , S 4 )-decomposable.Suppose, to the contrary of the conclusion, that there exists a (C 4 , S 4 )-decomposition D of K 5,5 + 3{a 0 b 0 }.Since there are at most two star with center a 0 (or b 0 ) and each edge joining a 0 and b 0 lies in exactly one subgraph in D, there are exactly three possibilities for the edges joining a 0 and b 0 to lie in the decomposition: in four 4-cycles, in three 4-cycles and a 4-star, or in two 4-cycles and two 4-stars.Let G 1 be the graph obtained from K 5,5 + 3{a 0 b 0 } by deleting the edges of four 4-cycles, and let G 2 be the graph obtained from K 5,5 + 3{a 0 b 0 } by deleting the edges of three 4-cycles or deleting the edges of two 4-cycles.Note that deg G1 x = 3 for x / ∈ {a 0 , b 0 }, which implies that there is no 4-star in G 1 .Since deg G2 x ≤ 3 for x ∈ {a 0 , b 0 }, there is no 4-star with center at a 0 or b 0 in G 2 .This leads to a contradiction and completes the proof. 2 We summarize the results discussed above in Table 1.
Tab Proof: The proof is divided into four parts according to the value of r.

Proposition 2 . 1 (
Hoffman [8]) For a positive integer k, a multigraph H has an S k -decomposition with central function c if and only if

2 . 3 . 3
and k divides 2mn.Removing the directions from the arcs of directed cycles in a − → C k -decomposition of K * m,n , we obtain the following result by Proposition 3.Lemma For positive integers m, n, and k, the multigraph 2K m,n is C k -decomposable if k is even, k ≥ 4, min{m, n} ≥ k/2, and k divides 2mn.Lemma 3.4 Let λ, k, m, and n be positive integers with λm ≡ λn ≡ k ≡ 0 (mod 2) and min{m r .By Lemma 3.4, λK k,k has a C k -decomposition D 1 .Trivially, λK k,r and λK r,k have S k -decompositions D 2 and D 3 , respectively.Thus 3 i=1 D i is a (C k , S k )-packing of λK n,n with leave λK r,r , as desired.Now we give the required covering.Let s = λr 2 .Let A 0 = {a 0 , a 1 , . . ., a (s−1)/2 }, A 1 = A − A 0 , B 0 = {b 0 , b 1 , . . ., b k−1 } and B 1 = B − B 0 .Define a k-cycle C and a (k − s + 1)-path P as follows:

Theorem 3 . 11
If λ and n are positive integers and k is a positive even integer with 4 ≤ k ≤ n, then p(λK n,n ; C k , S k ) = λn 2 /k and c(λK n,n ; C k , S k ) = λn 2 /k .