Multidesigns for the graph pair formed by the 6-cycle and 3-prism

Given two graphs G and H , a (G,H)-multidecomposition of Kn is a partition of the edges of Kn into copies of G and H such that at least one copy of each is used. We give necessary and sufficient conditions for the existence of (C6, C6)-multidecomposition of Kn where C6 denotes a cycle of length 6 and C6 denotes the complement of C6. We also characterize the cardinalities of leaves and paddings of maximum (C6, C6)-multipackings and minimum (C6, C6)-multicoverings, respectively.


Introduction
Let G and H be graphs. Denote the vertex set of G by V (G) and the edge set of G by E(G). A G-decomposition of H is a partition of E(H) into a set of edge-disjoint subgraphs of H each of which is isomorphic to G. Graph decompositions have been extensively studied. This is particularly true for the case where H ∼ = K n , see [2] for a recent survey. A G-decomposition of K n is sometimes referred to as a G-design of order n. As an extension of a graph decomposition we can permit more than one graph, up to isomorphism, to appear in the partition. A (G, H)multidecomposition of K n is a partition of E(K n ) into a set of edge-disjoint subgraphs each of which is isomorphic to either G or H, and at least one copy of G and one copy of H are elements of the partition. When a (G, H)-multidecomposition of K n does not exist, we would like to know how "close" we can get. More specifically, define a (G, H)-multipacking of K n to be a collection of edge-disjoint subgraphs of K n each of which is isomorphic to either G or H such that at least one copy of each is present. The set of edges in K n that are not used as copies of either G or H in the (G, H)-multipacking is called the leave of the (G, H)-multipacking. Similarly, define a (G, H)-multicovering of K n to be a partition of the multiset of edges formed by E(K n ) where some edges may be repeated into edge-disjoint copies of G and H such that at least one copy of each is present. The multiset of repeated edges is called the padding. A (G, H)-multipacking is called maximum if its leave is of minimum cardinality, and a (G, H)-multicovering is called minimum if its padding is of minimum cardinality. The term multidesign is used to encompass multidecompositions, multipackings, and multicoverings.
A natural way to form a pair of graphs is to use a graph and its complement. To this end, we have the following definition which first appeared in [1]. Let G and H be edge-disjoint, nonisomorphic, spanning subgraphs of K n each with no isolated vertices. We call (G, H) a graph pair of order n if E(G) ∪ E(H) = E(K n ). For example, the only graph pair of order 4 is (C 4 , E 2 ), where E 2 denotes the graph consisting of two disjoint edges. Furthermore, there are exactly 5 graph pairs of order 5. In this paper we are interested in the graph pair formed by a 6-cycle, denoted C 6 , and the complement of a 6-cycle, denoted C 6 .
Necessary and sufficient conditions for multidecompositions of complete graphs into all graph pairs of orders 4 and 5 were characterized in [1]. They also characterized the cardinalities of leaves and paddings of multipackings and multicoverings for the same graph pairs. We advance those results by solving the same problems for a graph pair of order 6, namely (C 6 , C 6 ). Note that C 6 is sometimes referred to as the 3-prism, but we used the former notation for brevity. We first address multidecompositions, then multipackings and multicoverings. Our main results are stated in the following three theorems.
Theorem 1.1. The complete graph K n admits a (C 6 , C 6 )-multidecomposition of K n if and only if n ≡ 0, 1 (mod 3) with n ≥ 6, except n ∈ {7, 9, 10}. Theorem 1.2. For each n ≡ 2 (mod 3) with n ≥ 8, a maximum (C 6 , C 6 )-multipacking of K n has a leave of cardinality 1. Furthermore, a maximum (C 6 , C 6 )-multipacking of K 7 has a leave of cardinality 6, and a maximum (C 6 , C 6 )-multipacking of either K 9 or K 10 has a leave of cardinality 3. Theorem 1.3. For each n ≡ 2 (mod 3) with n ≥ 8, a minimum (C 6 , C 6 )-multicovering of K n has a padding of cardinality 2. Furthermore, a minimum (C 6 , C 6 )-multicovering of K 7 has a padding of cardinality 6, and a minimum (C 6 , C 6 )-multicoveirng of either K 9 or K 10 has a padding of cardinality 2.
Let G and H be vertex-disjoint graphs. The join of G and H, denoted G∨H, is defined to be the graph with vertex set V (G)∪V (H) and edge set E(G)∪E(H)∪{{u, v} : u ∈ V (G), v ∈ V (H)}. We use the shorthand notation t i=1 G i to denote G 1 ∨ G 2 ∨ · · · ∨ G t , and when G i ∼ = G for all  Next, we state some known results on graph decompositions that will help us prove our main result. Sotteau's theorem gives necessary and sufficient conditions for complete bipartite graphs (denoted by K m,n when the partite sets have cardinalities m and n) to decompose into even cycles of fixed length. Here we state the result only for cycle length 6. Theorem 1.4 (Sotteau [5]). A C 6 -decomposition of K m,n exists if and only if m ≥ 4, n ≥ 4, m and n are both even, and 6 divides mn.
Another celebrated result in the field of graph decompositions is that the necessary conditions for a C k -decomposition of K n are also sufficient. Here we state the result only for k = 6. Theorem 1.5 (Šajna [4]). Let n be a positive integer. A C 6 -decomposition of K n exists if and only if n ≡ 1, 9 (mod 12).
The necessary and sufficient conditions for a C 6 -decomposition of K n are also known, and stated in the following theorem. Theorem 1.6 (Kang et al. [3]). Let n be a positive integer. A C 6 -decomposition of K n exists if and only if n ≡ 1 (mod 9).

Multidecompositions
We first establish the necessary conditions for a (C 6 , C 6 )-multidecomposition of K n .
Proof. Assume that a (C 6 , C 6 )-multidecomposition of K n exists. It is clear that condition (1) holds. Considering that the edges of K n are partitioned into subgraphs isomorphic to C 6 and C 6 , we have that there exist positive integers x and y such that n 2 = 6x + 9y. Hence, 3 divides n 2 , which implies n ≡ 0, 1 (mod 3), and condition (2) follows.

Small examples of multidecompositions
In this section we present various non-existence and existence results for (C 6 , C 6 )-multidecompositions of small orders. The existence results will help with our general constructions.

Non-existence results
The necessary conditions for the existence of a (C 6 , C 6 )-multidecomposition of K n fail to be sufficient in exactly three cases, namely n = 7, 9, 10. We will now establish the non-existence of (C 6 , C 6 )-multidecompositions of K n for these cases.
Proof. Assume the existence of a (C 6 , C 6 )-multidecomposition of K 7 , call it G. There must exist positive integers x and y such that 7 2 = 21 = 6x + 9y. The only solution to this equation is (x, y) = (2, 1); therefore, G must contain exactly one copy of C 6 . However, upon examing the degree of each vertex contained in the single copy of C 6 we see that there must exist a non-negative integer p such that 6 = 2p + 3. This is a contradiction. Thus, a (C 6 , C 6 )-multidecomposition of K 7 cannot exist. Lemma 2.3. A (C 6 , C 6 )-multidecomposition of K 9 does not exist.
Proof. Assume the existence of a (C 6 , C 6 )-multidecomposition of K 9 , call it G. There must exist positive integers x and y such that 9 2 = 36 = 6x + 9y. The only solution to this equation is (x, y) = (3, 2); therefore, G must contain exactly two copies of C 6 .
Turning to the degrees of the vertices in K 9 , we have that there must exist positive integers p and q such that 8 = 2p + 3q. The only possibilities are (p, q) ∈ {(4, 0), (1, 2)}. Note that K 6 does not contain two edge-disjoint copies of C 6 . Since G contains exactly two copies of C 6 , there must exist at least one vertex a ∈ V (K 9 ) that is contained in exactly one copy of C 6 . However, this contradicts the fact that vertex a must be contained in either 0 or 2 copies of C 6 . Thus, a (C 6 , C 6 )-multidecomposition of K 9 cannot exist.
Proof. Assume the existence of a (C 6 , C 6 )-multidecomposition of K 10 , call it G. There must exist positive integers x and y such that 10 2 = 45 = 6x + 9y. Thus, (x, y) ∈ {(6, 1), (3, 3)}; therefore, G must contain at least one copy of C 6 . However, if G consists of exactly one copy of C 6 , then the vertices of K 10 which are not included in this copy would have odd degrees remaining after the removal of the copy of C 6 . Thus, the case where (x, y) = (6, 1) is impossible.
Upon examining the degree of each vertex in K 10 , we see that there must exist positive integers p and q such that 9 = 2p + 3q. The only solutions to this equation are (p, q) ∈ {(3, 1), (0, 3)}. From the above argument, we know that G contains exactly 3 copies of C 6 , say A, B, and C. Let It must be the case that |X| ≥ 2 since K 10 has 10 vertices. It also must be the case that |X| ≤ 5 since K 6 does not contain two copies of C 6 . If |X| ∈ {2, 3}, then V (C) ∩ (V (A) V (B)) = ∅, where denotes the symmetric difference. This implies that there exists a vertex in V (K n ) that is contained in exactly 2 copies of C 6 in G, which is a contradiction.
Observe that any set consisting of either 4 or 5 vertices in C 6 must induce at least 3 or 6 edges, respectively. Furthermore, X ⊆ V (C) due to the degree constraints put in place by the existence of G. If |X| = 4 or |X| = 5, then X must induce at least 9 or at least 18 edges, respectively. This is a contradiction in either case. Thus, no (C 6 , C 6 )-multidecomposition of K 10 exists.

Existence results
We now present some multidecompositions of small orders that will be useful for our general recursive constructions.
The following is a (C 6 , C 6 )-multidecomposition of K 19 .
Proof. Let n = 6x + 1 for some integer x ≥ 2. The proof breaks into two cases.
Each of the k copies of K 13 formed by K 1 ∨ K 12 admits a (C 6 , C 6 )-multidecomposition by Example 1. The remaining edges form edge-disjoint copies of K 12,12 , which admits a C 6 -decomposition by Theorem 1.4. Thus, we obtain the desired (C 6 , C 6 )-multidecomposition of K n . Case 2: x = 2k + 1 for some integer k ≥ 2. Notice that K 12k+7 ∼ = K 1 ∨ K 6 ∨ k i=1 K 12 . The single copy of K 19 formed by K 1 ∨K 6 ∨K 12 admits a (C 6 , C 6 )-multidecomposition by Example 3. The remaining k − 1 copies of K 13 formed by K 1 ∨ K 12 each admit a (C 6 , C 6 )-multidecomposition by Example 1. The remaining edges form edge-disjoint copies of either K 6,12 or K 12,12 . Both of these graphs admit C 6 -decompositions by Theorem 1.4. Thus, we obtain the desired (C 6 , C 6 )multidecomposition of K n . Lemma 2.7. If n ≡ 3 (mod 6) with n ≥ 15, then K n admits a (C 6 , C 6 ) -multidecomposition.
Proof. Let n = 6x + 3 for some integer x ≥ 2. The proof breaks into two cases.
The remainder of the proof is similar to the proof of Case 1 of Lemma 2.6 where the ingredients required are C 6 -decompositions of K 12,12 , and K 12,14 , as well as (C 6 , C 6 )-multidecompositions of K 13 and K 15 . Case 2: x = 2k + 1 for some integer k ≥ 1.
The remainder of the proof is similar to the proof of Case 2 of Lemma 2.6 where the ingredients required are C 6 -decompositions of K 9 (which exists by Theorem 1.5), K 8,12 , and K 12,12 , as well as a (C 6 , C 6 )-multidecomposition of K 13 . Lemma 2.8. If n ≡ 4 (mod 6) with n ≥ 16, then K n admits a (C 6 , C 6 ) -multidecomposition.
The remainder of the proof is similar to the proof of Case 2 of Lemma 2.6 where the ingredients required are C 6 -decompositions of K 6,6 and K 6,10 , a C 6 -decomposition of K 10 (which exists by Theorem 1.6), as well as a (C 6 ,C 6 )-multidecomposition of K 6 .

Maximum Multipackings
Now we turn our attention to (C 6 , C 6 )-multipackings in the cases where (C 6 , C 6 )multidecompositions do not exist.

Small examples of maximum multipackings
Example 4. A maximum (C 6 , C 6 )-multipacking of K 7 has a leave of cardinality 6.
A (C 6 , C 6 )-multipacking of K 10 with a leave of cardinality 3 can be obtained by starting with a C 6 -decomposition of K 10 . Then remove three vertex-disjoint edges from one copy of C 6 , forming a C 6 . This gives us the desired (C 6 , C 6 )-multipacking of K 10 where the three removed edges form the leave.
Proof. Let n = 6x + 2 for some integer . Each of the x copies of K 8 formed by K 2 ∨ K 6 admit a (C 6 , C 6 )-multipacking with leave cardinality 1 by Example 5. Note that we can always choose the leave edge to be {u, v} in each of these multipackings. The remaining edges form edge disjoint copies of K 6,6 , each of which admits a C 6 -decomposition by Theorem 1.4. Thus, we obtain the desired (C 6 , C 6 )multipacking of K n . Lemma 3.2. If n ≡ 5 (mod 6) with n ≥ 11, then K n admits a (C 6 , C 6 )-multipacking with leave cardinality 1.
Proof. Let n = 6x + 5 for some integer x ≥ 1. Case 1: x = 2k for some integer k ≥ 1. Notice that K 12k+5 ∼ = K 1 ∨ K 16 ∨ k−1 i=1 K 12 . Each of the k − 1 copies of K 13 formed by K 1 ∨ K 12 admit a (C 6 , C 6 )-multidecomposition by Example 1. The copy of K 17 formed by K 1 ∨ K 16 admits a (C 6 , C 6 )-multipacking with leave of cardinality 1 by Example 9. The remaining edges form edge disjoint copies of K 12,12 or K 12,16 , each of which admits a C 6 -decomposition by Theorem 1.4. Thus, we obtain the desired (C 6 , C 6 )-multipacking of K n . Case 2: x = 2k + 1 for some integer k ≥ 1. Notice that K 12k+11 ∼ = K 1 ∨ K 10 ∨ k i=1 K 12 . On each of the k copies of K 13 formed by K 1 ∨ K 12 admit a (C 6 , C 6 )-multidecomposition by Example 1. The copy of K 11 formed by K 1 ∨ K 10 admits a (C 6 , C 6 )-multipacking with leave of cardinality 1 by Example 8. The remaining edges form edge disjoint copies of K 12,12 or K 10,12 , each of which admits a C 6 -decomposition by Theorem 1.4. Thus, we obtain the desired (C 6 , C 6 )-multipacking of K n .

Minimum Multicoverings
Now we turn our attention to minimum (C 6 , C 6 )-multicoverings in the cases where (C 6 , C 6 )multidecompositions do not exist.

Small examples of minimum multicoverings
Example 10. A minimum (C 6 , C 6 )-multicovering of K 7 has a padding of cardinality 6.
We first rule out the possibility of a minimum (C 6 , C 6 )-multicovering of K 7 with a padding of cardinality 3. The only positive integer solution to the equation 24 = 6x + 9y is (x, y) = (1, 2). In such a covering there would be one vertex left out of one of the copies of C 6 . It would be impossible to use all edges at this vertex with the remaining copies of C 6 and C 6 . Thus, the best possible cardinality of a padding is 6. Let V (K 7 ) = {1, ..., 7}. The following is a minimum (C 6 , C 6 )-multicovering of Example 12. A minimum (C 6 , C 6 )-multicovering of K 9 has a padding of cardinality 3.
A (C 6 , C 6 )-multicovering of K 9 with a padding of cardinality 3 can be obtained by starting with a C 6 -decomposition of K 9 which exists by Theorem 1.5. One of the copies of C 6 contained in this decomposition can be transformed into a copy of C 6 by carefully adding 3 edges. This gives us the desired (C 6 , C 6 )-multicovering of K 9 where the three added edges form the padding.
A (C 6 , C 6 )-multicovering of K 10 with a padding of cardinality 3 can be obtained by starting with a C 6 -decomposition of K 10 . One copy of C 6 can be transformed into two copies of C 6 by carefully adding three edges. This gives us the desired (C 6 , C 6 )-multicovering of K 10 where the three added edges form the padding.
Let V (K 17 ) = {1, ..., 17}. Apply Theorem 1.5 and let B 1 be a C 6 -decomposition on the copy of K 9 formed by the subgraph induced by the vertices {9, . . . , 17}. Apply Theorem 1.4 and let B 2 be a C 6 -decomposition of the copy of K 6,8 formed by the subgraph of K 17 with vertex bipartition Lemma 4.2. If n ≡ 5 (mod 6) with n ≥ 11, then K n admits a minimum (C 6 , C 6 )multicovering with a padding of cardinality 2.
Proof. Let n = 6x + 5 for some integer x ≥ 1. The proof breaks into two cases. Case 1: x = 2k for some integer k ≥ 1. Notice that K 12k+5 ∼ = K 1 ∨ K 4 ∨ k i=1 K 12 . One copy of K 17 is formed by K 1 ∨ K 4 ∨ K 12 , and admits a (C 6 , C 6 )-multicovering with a padding of cardinality 2 by Example 15. The k − 1 copies of K 13 formed by K 1 ∨ K 12 admit a (C 6 , C 6 )multidecomposition by Example 1. The remaining edges form edge disjoint copies of K 12,12 or K 4,12 , each of which admits a C 6 -decomposition by Theorem 1.4. Thus, we obtain the desired (C 6 , C 6 )-multicovering of K n . Case 2: x = 2k + 1 for some integer k ≥ 1. Notice that K 12k+11 ∼ = K 1 ∨ K 4 ∨ K 6 ∨ k i=1 K 12 . One copy of K 11 is formed by K 1 ∨ K 4 ∨ K 6 , and admits a (C 6 , C 6 )-multicovering with a padding of cardinality 2 by Example 14. The k copies of K 13 formed by K 1 ∨ K 12 admit a (C 6 , C 6 )multidecomposition by Example 1. The remaining edges form edge disjoint copies of K 12,12 , K 4,12 , or K 6,12 , each of which admits a C 6 -decomposition by Theorem 1.4. Thus, we obtain the desired (C 6 , C 6 )-multicovering of K n .

Conclusion
The cardinalities of the leaves of maximum (C 6 , C 6 )-multipackings and paddings of minimum (C 6 , C 6 )-multicoverings of K n have been characterized. However, the achievable structures of these leaves and paddings are still yet to be characterized. This leads to the following open question.
Open Problem 1. For each positive integer n, characterize all possible graphs (multigraphs) which are leaves (paddings) of a (C 6 , C 6 )-multipacking (multicovering) of K n .