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Hung- Chih Lee Department of Information Technology Ling Tung University

Maximum packings and minimum coverings of multigraphs with paths and stars. Hung- Chih Lee Department of Information Technology Ling Tung University. Chun-Cheng Chen Department of Mathematics National Central University. Outline. Introduction. Packing and covering of λ K n.

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Hung- Chih Lee Department of Information Technology Ling Tung University

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  1. Maximum packings and minimum coverings of multigraphswith paths and stars Hung-ChihLee Department of Information Technology Ling Tung University Chun-Cheng Chen Department of Mathematics National Central University

  2. Outline Introduction Packing and covering of λKn Packing and covering of λKn,n

  3. Introduction Kn : the complete graph with nvertices. Km,n : the complete bipartite graph with parts of sizes m and n. If m = n, the complete bipartite graph is referred to as balanced. Sk(k-star) : the complete bipartite graph K1,k. Pk(k-path) : a path on k vertices. Ck(k-cycle) : a cycle of length k.

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

  5. Example. K6 K3,4 P4|K6 S3|K3,4

  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.

  7. Example. K6 K3,4 (P4, S3)|K6 (P4, S3)|K3,4

  8. Abueida and Daven[3] investigated the problem of (Kk,Sk)-decomposition of the complete graph Kn. [3] A. Abueida and M. Daven, Multidecompositons of the complete graph, Ars Combin. 72(2004), 17-22 . Abueida and Daven[4] investigated the problem of the (C4, E2)-decomposition of several graph products where E2 denotes two vertex disjoint edges. [4] A. Abueida and M. Daven, Multidecompositions of several graph products, Graphs Combin. 29 (2013), 315-326. Abueidaand O‘Neil[7] settled the existence problem for (Ck, Sk-1)-decomposition of the complete multigraph λKnfor k∈{3,4,5}. [7] A. Abueida and T. O'Neil, Multidecomposition of λKminto small cycles and claws, Bull. Inst. Combin. Appl. 49 (2007), 32-40.

  9. Priyadharsini and Muthusamy[15, 16] gave necessary and sufficient conditions for the existence of (Gn,Hn)-decompositions of λKnand λKn,n where Gn,Hn∈{Cn,Pn,Sn-1}. [15] H. M. Priyadharsini and A. Muthusamy, (Gm,Hm)-multifactorization of λKm, J. Com-bin. Math. Combin. Comput. 69 (2009), 145-150. [16] H. M. Priyadharsini and A. Muthusamy, (Gm,Hm)-multidecomposition of Km,m(λ),Bull. Inst. Combin. Appl. 66 (2012), 42-48.

  10. 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 GH is isomorphic to Km. Abueidaand Daven[2] and Abueida, Daven and Roblee[5] completely determined the values of nfor which λKnadmits 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. [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.

  11. Abueida, Clark and Leach [1] and Abueida and Hampson[6] considered the existence of decompositions of Kn−Ffor 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, ArsCombin. 93 (2009), 403-407. [6] A. Abueida and C. Hampson, Multidecomposition of Kn − F into graph-pairs of order 5 where F is a Hamilton cycle or an (almost) 1-factor, ArsCombin. 97 (2010), 399-416 Shyu[17] investigated the problem of decomposing Kn 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.

  12. In [18, 19], Shyu considered the existence of a decomposition of Kn 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, ArsCombin. 107 (2012), 209-224. Shyu[20] investigated the problem of decomposing Kn into cycles and stars with kedges, settling the case k = 4. [20] T.-W. Shyu, Decomposition of complete graphs into cycles and stars, Graphs Combin. 29 (2013), 301-313.

  13. In [21], Shyu considered the existence of a decomposition of Km,ninto 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. Lee[12] and Lee and Lin[13] established necessary and sufficient conditions for the existence of (Ck,Sk)-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, ArsCombin. 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. When a multigraph H does not admit an (F,G)-decomposition, two natural questions arise : 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 Hso 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.

  15. Let F, G, and H be multigraphs. For subgraphsL 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 cardinalityis 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.

  16. Example. (P5, S4)|K6–E(P4) K6 (P5, S4)|K6+E(P2)

  17. Abueida and Daven[3] obtained the maximum packing and the minimum covering of the complete graph Kn with (Kk,Sk). [3] A. Abueida and M. Daven, Multidecompositons of the complete graph, ArsCombin. 72(2004), 17-22 . Abueida and Daven[2] and Abueida, Daven and Roblee[5] gave the maximum packing and the minimum covering of Kn and λKn 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. [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.

  18. Packing and covering of λKn Let G be a multigraph. The degree of a vertex x of G (degG(x)) : the number of edges incident with x. The center of Sk: the vertex of degree k in Sk. The endvertexof Sk : the vertex of degree 1 in Sk. (x;y1,y2,…,yk) : the Skwith center x and endverticesy1,y2,…,yk.

  19. v1v2… vk: the Pkthrough vertices v1,v2,…,vk in order, and the vertices v1 and vk are referred to as its origin and terminus. If P = x1x2…xt, Q = y1y2…ys and xt = y1, then P + Q = x1x2…xty2…ys. Pk(v1,vk) : a Pkwith origin v1 and terminus vk.

  20. G[U] : the subgraph of G induced by U. G[U,W] : the maximal bipartite subgraph of G with bipartition (U,W). When G1,G2,…,Gtare multigraphs, not necessarily disjoint. G1G2…Gtor : the graph with vertex set and edge set . When the edge sets are disjoint, G =expresses the decomposition of G into G1,G2,…,Gt.

  21. Proposition 2.1. For positive integers λ, n, and t, and any sequence m1,m2,…,mt of positive integers, the complete multigraphλKncan be decomposed into paths of lengths m1,m2,…,mtif and only if mi ≤ n-1 and m1+m2+…+mt= |E(λKn)|. [9] Darryn Bryant, Packing paths in complete graphs, J. Combin. Theory Ser. B 100 (2010), 206-215.

  22. Proposition 2.2. For a positive even integer n and V (Kn) = {x0, x1,…, xn-1}, the complete graph Kn can be decomposed into copies of n-paths Pn(x0,x), Pn(x1,x1+ ) ,…,Pn(x1, xn-1). [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.

  23. 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(Kn)| ≡ t (mod k), then Knhas a (Pk+1, Sk)-packing with leave Pt+1. Proof. Let n = qk+r and |E(Kn)| = pk+t with integers q,p,rand 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.

  24. Case 2. m is even. Proposition 2.2. Lemma 2.4. For a positive even integer n and V (Kn) = {x0,x1,…,xn-1}, the complete graph Kn can be decomposed into copies of n-paths Pn(x0,x), Pn(x1,x1+) ,…,Pn(x1,xn-1). Let k, m, p, and s be positive integers and let t be a nonnegative integer with max{m,t} ≤ k-1. If then p-(s+1)m > 0. Kn= Km+sk+1 = KmKm,sk+1Ksk+1. Proposition 2.1. By Proposition 2.2, Km=Pm. For positive integers λ, n, and t, and any sequence m1,m2,…,mt of positive integers, the complete multigraphλKncan be decomposed into paths of lengths m1,m2,…,mtif and only if mi ≤ n-1 and m1+ m2+…+mt= |E(λKn)|. By Proposition 2.1 and Lemma 2.4, Ksk+1=Pk-m(p-ms-)Pk+1Pt+1. P … Ksk+1 Sk|K1,sk Pk+1 Sk|K1,sk … Km … P Hence, Kn= msSk(p-ms)Pk+1Pt+1.

  25. Theorem 2.6. 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(λKn)| ≡t (mod k), then λKnhas a (Pk+1, Sk)-packing with leave Pt+1.

  26. Corollary 2.7. For positive integers λ, k, and n with k ≥ 3, the complete multigraphλKnis (Pk+1,Sk)-decomposable if and only if n ≥ k + 1, ≡ 0 (mod k) and (λ,n) ≠(1, k + 1).

  27. For positive integers k, n, and t with k≤n-1 and t ≤ k-1. If λKnhas a (Pk+1,Sk)-packing with leave Pt+1, then it has a (Pk+1, Sk)-covering with padding Pk-t+1. Since p(λKn; Pk+1,Sk) ≤≤ ≤ c(λKn; Pk+1,Sk), we have the following by Theorem 2.6. Theorem 2.8. If λ, n, and k are positive integers with n ≥ k+2 ≥ 5, then p(λKn; Pk+1,Sk) =and c(λKn; Pk+1, Sk) =.

  28. Packing and covering of λKn,n (v1,v2,…,vk) : the k-cycle of G through vertices v1,v2,…,vk 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 Skin the decomposition whose center is v.

  29. Proposition 3.1. For a positive integer k, a multigraph H has an Sk-decomposition with central function c if and only if let c : V → N{0}. (i) ; (ii) for all ; (iii) for all , . where denotes the number of edges of H with both ends in S. [11] D.G. Hoffman, The real truth about star designs, Discrete Math. 284 (2004), 177-180.

  30. Lemma 3.4. Let λ, k, and n be positive integers with 3 ≤ k < n < 2k. If , then λKn,nhas a (Pk+1,Sk)-packing P with |P|=and a (Pk+1,Sk)-covering C with |C|=. Lemma 3.6. Let λ, k, and n be positive integers with 3 ≤ k < n < 2k. If , then λKn,nhas a (Pk+1,Sk)-packing P with |P|=and a (Pk+1,Sk)-covering C with |C|=.

  31. Proof of Lemma 3.4. Let n = k + r. The assumption k < n < 2k implies 0 < r < k. Packing. Lemma 3.3. If λand k are positive integers with k ≥ 2, then λKk,kis (Pk+1,Sk)-decomposable. leave λKn,n=λKk,kλKk,r λKr,kλKr,r By Lemma 3.3.⇒(Pk+1, Sk) |Kk,k

  32. Covering Claim. λKn,n+E(Pk-s+1)=Pk+1Sk. Let s =. Let , , and Define a (k + 1)-path Pas follows: P Let P’ be the (k-s+1)-subpath of Pwith origin for odd k and with origin for even k. … … … … … … … … s is odd. k is odd. H = λKn,n - E(P) + E(P’). … … … … … … … … … … … … … … … … P P’

  33. Proposition 3.1. For a positive integer k, a multigraph H has an Sk-decomposition with central function c if and only if let c : V → N{0}. (i) ; (ii) for all ; (iii) for all , . where denotes the number of edges of H with both ends in S. Claim.Sk|H Define a function c : V (H)→N{0}as follows :

  34. Theorem 3.7. If λ, k, and n are positive integers with 3 ≤k ≤n, then p(λKn,n; Pk+1, Sk) = and c(λKn,n; Pk+1, Sk) = . Corollary 3.8. For positive integers λ, k and n with k≥3, the balanced complete bipartite multigraphλKn,nis (Pk+1,Sk)-decomposable if and only if k ≤n and is divisible by k.

  35. [1] A. Abueida, S. Clark, and D. Leach, Multidecomposition of the complete graph into graph pairs of order 4 with various leaves, ArsCombin. 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, ArsCombin. 72(2004), 17-22 . [4] A. Abueida and M. Daven, Multidecompositions of several graph products, Graphs Combin. 29 (2013), 315-326. [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 Kn − F into graph-pairs of order 5 where F is a Hamilton cycle or an (almost) 1-factor, ArsCombin. 97 (2010), 399-416 [7] A. Abueida and T. O'Neil, Multidecomposition of λKminto small cycles and claws, Bull. Inst. Combin. Appl. 49 (2007), 32-40. [8] J. Bosák, Decompositions of Graphs, Kluwer, Dordrecht, Netherlands, 1990.

  36. [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, ArsCombin. 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. [15] H. M. Priyadharsini and A. Muthusamy, (Gm,Hm)-multifactorization of λKm, J. Com-bin. Math. Combin. Comput. 69 (2009), 145-150. [16] H. M. Priyadharsini and A. Muthusamy, (Gm,Hm)-multidecomposition of Km,m(λ),Bull. Inst. Combin. Appl. 66 (2012), 42-48.

  37. [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, ArsCombin. 97(2010), 257-270. [19] T.-W. Shyu, Decomposition of complete graphs into paths of length three and triangles, ArsCombin. 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.

  38. Thank you for your listening.

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