Download presentation

Presentation is loading. Please wait.

Published byMelany Rule Modified over 3 years ago

1
Chun-Cheng Chen Department of Mathematics National Central University 1

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

3
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.

4
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.

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

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

8
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.

9
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.

10
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.

11
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

12
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

13
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

14
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.

15
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.

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

17
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.

18
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.

19
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

20
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.

21
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.

22
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 )|.

23
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.

24
24 Lemma 2.3. Lemma 2.4.

25
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.

26
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 )|.

27
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.

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

29
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 )|.

30
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 )|.

31
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.

32
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.

33
33 Corollary 2.7.

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

35
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.

36
36 Case 1. λ is even.

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

38
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.

39
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.

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

41
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.

42
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.

43
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.

44
44 Lemma 3.4. Lemma 3.6.

45
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.

46
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.

47
47 Proposition 3.1. Claim. S k |H

48
48 Theorem 3.7. Corollary 3.8.

49
[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.

50
[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.

51
[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.

52
52 Thank you for your listening.

Similar presentations

OK

Lecture 5 Graph Theory. Graphs Graphs are the most useful model with computer science such as logical design, formal languages, communication network,

Lecture 5 Graph Theory. Graphs Graphs are the most useful model with computer science such as logical design, formal languages, communication network,

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google