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Chun-Cheng Chen Department of Mathematics National Central University 1

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Outline Introduction Packing and covering of λK n Packing and covering of λK n,n 2

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Introduction 3 K n : the complete graph with n vertices. K m,n : the complete bipartite graph with parts of sizes m and n. If m = n, the complete bipartite graph is referred to as balanced. S k (k-star) : the complete bipartite graph K 1,k. P k (k-path) : a path on k vertices. C k (k-cycle) : a cycle of length k.

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4 λ λ λ H : the multigraph obtained from H by replacing each edge e by λ edges each having the same endpoints as e. λ When λ = 1, 1H is simply written as H. A decomposition of H : a set of edge-disjoint subgraphs of H whose union is H. A G-decomposition of H : a decomposition of H in which each subgraph is isomorphic to G. If H has a G-decomposition, we say that H is G- decomposable and write G|H.

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5 Example. P 4 |K 6 K6K6 K 3,4 S 3 |K 3,4

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6 An (F, G)-decomposition of H : 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.

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7 Example. (P 4, S 3 )|K 6 K6K6 K 3,4 (P 4, S 3 )|K 3,4

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8 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. [4] A. Abueida and M. Daven, Multidecompositions of several graph products, Graphs Combin. 29 (2013), 315-326. Abueida and Daven [3] investigated the problem of (K k, S k )-decomposition of the complete graph K n. [3] A. Abueida and M. Daven, Multidecompositons of the complete graph, Ars Combin. 72(2004), 17-22. λ Abueida and O‘Neil [7] settled the existence problem for (C k, S k-1 )-decomposition of the complete multigraph λ K n for k ∈ {3,4,5}. λ [7] A. Abueida and T. O'Neil, Multidecomposition of λ K m into small cycles and claws, Bull. Inst. Combin. Appl. 49 (2007), 32-40.

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9 λλ Priyadharsini and Muthusamy [15, 16] gave necessary and sufficient conditions for the existence of (G n,H n )-decompositions of λ K n and λ K n,n where G n,H n ∈ {C n,P n,S n-1 }. λ [15] H. M. Priyadharsini and A. Muthusamy, (G m,H m )-multifactorization of λ K m, J. Com-bin. Math. Combin. Comput. 69 (2009), 145-150. [16] H. M. Priyadharsini and A. Muthusamy, (G m,H m )-multidecomposition of λ K m,m ( λ ),Bull. Inst. Combin. Appl. 66 (2012), 42-48.

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10 λ Abueida and Daven [2] and Abueida, Daven and Roblee [5] completely determined the values of n for which λ K n admits a (G,H)-decomposition where (G,H) is a graph-pair of order 4 or 5. [2] A. Abueida and M. Daven, Multidesigns for graph-pairs of order 4 and 5, Graphs Com-bin. 19 (2003), 433-447. λ [5] A. Abueida, M. Daven, and K. J. Roblee, Multidesigns of the λ -fold complete graph for graph-pairs of order 4 and 5, Australas J. Combin. 32 (2005), 125-136.

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11 Shyu [17] investigated the problem of decomposing K n into paths and stars with k edges, giving a necessary and sufficient condition for k = 3. [17] T.-W. Shyu, Decomposition of complete graphs into paths and stars, Discrete Math. 310 (2010), 2164-2169. 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. [1] A. Abueida, S. Clark, and D. Leach, Multidecomposition of the complete graph into graph pairs of order 4 with various leaves, Ars Combin. 93 (2009), 403-407. − [6] A. Abueida and C. Hampson, Multidecomposition of K n − F into graph-pairs of order 5 where F is a Hamilton cycle or an (almost) 1-factor, Ars Combin. 97 (2010), 399-416

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Shyu [20] investigated the problem of decomposing K n into cycles and stars with k edges, settling the case k = 4. [20] T.-W. Shyu, Decomposition of complete graphs into cycles and stars, Graphs Combin. 29 (2013), 301-313. In [18, 19], 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}. [18] T.-W. Shyu, Decompositions of complete graphs into paths and cycles, Ars Combin. 97(2010), 257-270. [19] T.-W. Shyu, Decomposition of complete graphs into paths of length three and triangles, Ars Combin. 107 (2012), 209-224. 12

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Lee [12] and Lee and Lin [13] 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. [12] H.-C. Lee, Multidecompositions of complete bipartite graphs into cycles and stars, Ars Combin. 108 (2013), 355-364. [13] H.-C. Lee and J.-J. Lin, Decomposition of the complete bipartite graph with a 1- factor removed into cycles and stars, Discrete Math. 313 (2013), 2354-2358. In [21], 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. [21] T.-W. Shyu, Decomposition of complete bipartite graphs into paths and stars with same number of edges, Discrete Math. 313 (2013), 865-871. 13

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14 When a multigraph 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? (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? These questions are respectively called the maximum packing problem and the minimum covering problem of H with F and G.

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15 Let F, G, and H be multigraphs. For subgraphs L and R of H. − An (F,G)-packing ((F,G)-covering) of H with leave L (padding R) is an (F,G)-decomposition of H − E(L) (H+E(R)). An (F,G)-packing ((F,G)-covering) of H with the largest (smallest) cardinality is a maximum (F,G)-packing (minimum (F,G)-covering) of H, and its cardinality is the (F,G)-packing number ((F,G)-covering number) of H, denoted by p(H;F,G) (c(H;F,G)). An (F,G)-decomposition of H is a maximum (F,G)-packing (minimum (F,G)-covering) with leave (padding) the empty graph.

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16 Example. (P 5, S 4 )|K 6 –E(P 4 ) K6K6 (P 5, S 4 )|K 6 +E(P 2 )

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17 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. [2] A. Abueida and M. Daven, Multidesigns for graph-pairs of order 4 and 5, Graphs Com-bin. 19 (2003), 433-447. λ [5] A. Abueida, M. Daven, and K. J. Roblee, Multidesigns of the λ -fold complete graph for graph-pairs of order 4 and 5, Australas J. Combin. 32 (2005), 125-136. Abueida and Daven [3] obtained the maximum packing and the minimum covering of the complete graph K n with (K k,S k ). [3] A. Abueida and M. Daven, Multidecompositons of the complete graph, Ars Combin. 72(2004), 17-22.

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Packing and covering of λK n 18 The existence problem for (P k+1, S k )-decomposition of λK n are investigated. Determining the packing number and the covering number of K n with P k+1 and S k.

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19 Let G be a multigraph. The degree of a vertex x of G (deg G (x)) : the number of edges incident with x. The center of S k : the vertex of degree k in S k. The endvertex of S k : the vertex of degree 1 in S k. (x;y 1,y 2,…,y k ) : the S k with center x and endvertices y 1,y 2,…,y k. Packing and covering of λK n

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20 v 1 v 2 … v k : the P k through vertices v 1,v 2,…,v k in order, and the vertices v 1 and v k are referred to as its origin and terminus. If P = x 1 x 2 …x t, Q = y 1 y 2 …y s and x t = y 1, then P + Q = x 1 x 2 …x t y 2 …y s. P k (v 1,v k ) : a P k with origin v 1 and terminus v k.

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21 G[U] : the subgraph of G induced by U. G[U,W] : the maximal bipartite subgraph of G with bipartition (U,W). When G 1,G 2,…,G t are multigraphs, not necessarily disjoint.

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22 [9] Darryn Bryant, Packing paths in complete graphs, J. Combin. Theory Ser. B 100 (2010), 206-215. Proposition 2.1. λ λ ≤ For positive integers λ, n, and t, and any sequence m 1,m 2,…,m t of positive integers, the complete multigraph λ K n can be decomposed into paths of lengths m 1,m 2,…,m t if and only if m i ≤ n-1 and λ m 1 +m 2 +…+m t = |E( λ K n )|.

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23 [8] J. Bosák, Decompositions of Graphs, Kluwer, Dordrecht, Netherlands, 1990. [10] P. Hell and A. Rosa, Graph decompositions, handcuffed prisoners and balanced P-designs, Discrete Math. 2 (1972), 229-252. Proposition 2.2.

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24 Lemma 2.3. Lemma 2.4.

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25 Theorem 2.5. ≥ ≥ ≤ ≡ Let n and k be positive integers and let t be a nonnegative integer with k ≥ 3,n ≥ k + 2, and t ≤ k-1. If |E(K n )| ≡ t (mod k), then K n has a (P k+1, S k )-packing with leave P t+1. Proof. ≤ ≤ Let n = qk+r and |E(K n )| = pk+t with integers q,p,r and 0 ≤ r ≤ k -1. ≠ ≤ ≤ Now we consider the case r ≠ 1, we set n = m + sk + 1 where m and s are positive integers with 1 ≤ m ≤ k-1. Case 1. m is odd. Case 2. m is even.

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26 Proof. ≤ ≤ Let n = qk+r and |E(K n )| = pk+t with integers q,p,r and 0 ≤ r ≤ k -1. Note that |E(K qk )| = (p-q)k+t. Hence, K n has a (P k+1,S k )-packing with leave P t+1 for r = 1. Proposition 2.1. λ λ ≤ λ For positive integers λ, n, and t, and any sequence m 1,m 2,…, m t of positive integers, the complete multigraph λ K n can be decomposed into paths of lengths m 1,m 2,…,m t if and only if m i ≤ n-1 and m 1 + m 2 + … + m t =|E( λ K n )|.

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27 ≠ Now we consider the case r ≠ 1. ≥ ≥ 2. If r = 0, since n ≥ k + 2, so q ≥ 2. Moreover, n = qk = (k-1)+(q-1)k+1. ≤ ≤ Let n = qk+r and |E(K n )| = pk+t with integers q,p,r and 0 ≤ r ≤ k -1. ≠ ≤ ≤ Thus, for r ≠ 1 we set n = m + sk + 1 where m and s are positive integers with 1 ≤ m ≤ k-1. …,…, Let A = {x 0, x 1, …, x m-1 } and B = {y 0, y 1, …, y sk }. Case 1. m is odd. Case 2. m is even.

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28 Case 1. m is odd. K sk+1 x0x0…… PkPk sk+1 K1K1 S k |K 1,sk P k+1

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29 Case 1. m is odd. K sk+1 KmKm S k |K 1,sk… … P k+1 … S k |K 1,sk Lemma 2.3. Lemma 2.4. ≥ If m ≥ 3. Proposition 2.1. λ λ ≤ λ For positive integers λ, n, and t, and any sequence m 1,m 2,…, m t of positive integers, the complete multigraph λ K n can be decomposed into paths of lengths m 1,m 2,…, m t if and only if m i ≤ n-1 and m 1 + m 2 + … + m t = |E( λ K n )|.

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30 Case 2. m is even. K sk+1 KmKm S k |K 1,sk… P k+1 S k |K 1,sk… … Proposition 2.2. Lemma 2.4. Proposition 2.1. λ λ ≤ λ For positive integers λ, n, and t, and any sequence m 1,m 2,…,m t of positive integers, the complete multigraph λ K n can be decomposed into paths of lengths m 1,m 2,…,m t if and only if m i ≤ n-1 and m 1 + m 2 +…+m t = |E( λ K n )|.

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31 Theorem 2.6. λ ≥ ≥ ≤ λ ≡λ Let λ, n and k be positive integers and let t be a nonnegative integer with k ≥ 3, n ≥ k+2, and t ≤ k-1. If |E( λ K n )| ≡ t (mod k), then λ K n has a (P k+1, S k )-packing with leave P t+1.

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32 Proof. …, ≡ ≤ ≤ Let V (K n ) = {x 0,x 1, …, x n-1 } and |E(K n )| ≡ r (mod k) with 0 ≤ r ≤ k -1. λ ≡ λ ≡ From the assumption |E( λ K n )| ≡ t (mod k), we have λ r ≡ t (mod k). λλ …, λ- λ K n can be decomposed into λ copies K n 's, say G 0,G 1, …, G λ- 1. Theorem 2.5. ≥ ≥ ≤ ≡ Let n and k be positive integers and let t be a nonnegative integer with k ≥ 3,n ≥ k + 2, and t ≤ k-1. If |E(K n )| ≡ t (mod k), then K n has a (P k+1, S k )-packing with leave P t+1.

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33 Corollary 2.7.

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34 Proof. Since |V(K n )|=n and |V(P k+1 )|=k+1, n ≥ k + 1 is necessary. Necessity

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35 Proof. Suffciency λ We distinguish two cases according to the parity of λ. Theorem 2.6. λ ≥ ≥ ≤ λ ≡λ Let λ, n and k be positive integers and let t be a nonnegative integer with k ≥ 3, n ≥ k+2, and t ≤ k-1. If |E( λ K n )| ≡ t (mod k), then λ K n has a (P k+1, S k )-packing with leave P t+1.

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36 Case 1. λ is even.

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37 Case 2. λ is odd. 2K k+1 K k+1… Proposition 2.2. By case1. By Proposition 2.2. …

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38 ≤ ≤ For positive integers k, n, and t with k ≤ n-1 and t ≤ k-1. λ If λ K n has a (P k+1, S k )-packing with leave P t+1, then it has a (P k+1, S k )- covering with padding P k-t+1. Theorem 2.8.

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Packing and covering of λK n,n 39 (v 1,v 2,…,v k ) : the k-cycle of G through vertices v 1,v 2,…,v k in order. μ(uv) : the number of edges of G joining u and v. 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 S k in the decomposition whose center is v.

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40 Proposition 3.1. [11] D.G. Hoffman, The real truth about star designs, Discrete Math. 284 (2004), 177-180.

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41 Proposition 3.2. ≡ There exists a P k+1 -decomposition of K m,n if and only if mn ≡ 0(mod k) and one of the following cases occurs. CasekmnConditions 1even ≤≤ k ≤ 2m, k ≤ 2n, not both equalities 2even odd ≤≤ k ≤ 2m-2, k ≤ 2n 3evenoddeven ≤≤ k ≤ 2m, k ≤ 2n-2 4oddevenodd ≤≤ k ≤ 2m-1, k ≤ 2n-1 5oddevenodd ≤≤ k ≤ 2m-1, k ≤ n 6odd even ≤≤ k ≤ m, k ≤ 2n-1 7odd ≤≤ k ≤ m, k ≤ n [14] C. A. Parker, Complete bipartite graph path decompositions, Ph.D. Thesis, Auburn Uni-versity, Auburn, Alabama, 1998.

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42 Lemma 3.3. λ ≥ λ If λ and k are positive integers with k ≥ 2, then λ K k,k is (P k+1, S k )-decomposable.

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43 Lemma 3.3. λ ≥ λ If λ and k are positive integers with k ≥ 2, then λ K k,k is (P k+1, S k )-decomposable. Proof. By Proposition 3.2.

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44 Lemma 3.4. Lemma 3.6.

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45 Let n = k + r. The assumption k < n < 2k implies 0 < r < k. Proof of Lemma 3.4. Packing. By Lemma 3.3.⇒ (P k+1, S k ) |K k,k leave Lemma 3.3. λ ≥ λ If λ and k are positive integers with k ≥ 2, then λ K k,k is (P k+1, S k )- decomposable.

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46 Define a (k + 1)-path P as follows:………… ………… ………… ………… ………… ………… λ H = λ K n,n - E(P) + E(P’). P P’ Covering k is odd. s is odd.

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47 Proposition 3.1. Claim. S k |H

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48 Theorem 3.7. Corollary 3.8.

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[1] A. Abueida, S. Clark, and D. Leach, Multidecomposition of the complete graph into graph pairs of order 4 with various leaves, Ars Combin. 93 (2009), 403-407. [2] A. Abueida and M. Daven, Multidesigns for graph-pairs of order 4 and 5, Graphs Com-bin. 19 (2003), 433-447. [3] A. Abueida and M. Daven, Multidecompositons of the complete graph, Ars Combin. 72(2004), 17-22. [4] A. Abueida and M. Daven, Multidecompositions of several graph products, Graphs Combin. 29 (2013), 315-326. λ [5] A. Abueida, M. Daven, and K. J. Roblee, Multidesigns of the λ -fold complete graph for graph-pairs of order 4 and 5, Australas J. Combin. 32 (2005), 125-136. − [6] A. Abueida and C. Hampson, Multidecomposition of K n − F into graph-pairs of order 5 where F is a Hamilton cycle or an (almost) 1-factor, Ars Combin. 97 (2010), 399-416 λ [7] A. Abueida and T. O'Neil, Multidecomposition of λ K m into small cycles and claws, Bull. Inst. Combin. Appl. 49 (2007), 32-40. [8] J. Bosák, Decompositions of Graphs, Kluwer, Dordrecht, Netherlands, 1990.

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[9] Darryn Bryant, Packing paths in complete graphs, J. Combin. Theory Ser. B 100 (2010), 206-215. [10] P. Hell and A. Rosa, Graph decompositions, handcuffed prisoners and balanced P-designs, Discrete Math. 2 (1972), 229-252. [11] D.G. Hoffman, The real truth about star designs, Discrete Math. 284 (2004), 177-180. [12] H.-C. Lee, Multidecompositions of complete bipartite graphs into cycles and stars, Ars Combin. 108 (2013), 355-364. [13] H.-C. Lee and J.-J. Lin, Decomposition of the complete bipartite graph with a 1- factor removed into cycles and stars, Discrete Math. 313 (2013), 2354-2358. [14] C. A. Parker, Complete bipartite graph path decompositions, Ph.D. Thesis, Auburn Uni-versity, Auburn, Alabama, 1998. λ [15] H. M. Priyadharsini and A. Muthusamy, (G m,H m )-multifactorization of λ K m, J. Com-bin. Math. Combin. Comput. 69 (2009), 145-150. λ [16] H. M. Priyadharsini and A. Muthusamy, (G m,H m )-multidecomposition of K m,m ( λ ),Bull. Inst. Combin. Appl. 66 (2012), 42-48.

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[17] T.-W. Shyu, Decomposition of complete graphs into paths and stars, Discrete Math. 310 (2010), 2164-2169. [18] T.-W. Shyu, Decompositions of complete graphs into paths and cycles, Ars Combin. 97(2010), 257-270. [19] T.-W. Shyu, Decomposition of complete graphs into paths of length three and triangles, Ars Combin. 107 (2012), 209-224. [20] T.-W. Shyu, Decomposition of complete graphs into cycles and stars, Graphs Combin. 29 (2013), 301-313. [21] T.-W. Shyu, Decomposition of complete bipartite graphs into paths and stars with same number of edges, Discrete Math. 313 (2013), 865-871.

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52 Thank you for your listening.

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