圈分割的極圖

全文

(1)國立交通大學應用數學系碩士論文. 圈分割的極圖 Extremal Graphs of Ck-decomposition. 研究生：陳亮銓指導教授：傅恆霖. 教授. 中華民國九十四年六月.

(2) 圈分割的極圖研究生: 陳亮銓. 指導教授: 傅恆霖教授. 國立交通大學應用數學系. 摘要如果一個圖 G 的邊集合可以分成一些子集合的聯集，而每一個子集合都導出一個 k 圈，圖 G 就稱為有 k 圈分割。很明顯地，如果圖 G 有 k 圈分割，圖 G 一定是一個偶圖，而且 k 會整除圖 G 的邊數。我們稱一個滿足上面兩個條件的圖為 k 充分圖。不難發現，一個 k 充分圖可能沒有 k 圈分割。在論文中的第一部份，將探討一個有 n 個點，是 r 正則且 k 充分，但是卻不存在 k 圈分割的圖。利用直接建構法說明，r 是如何根據 k 和 n 的不同，得到不同的下界。第二部份，探討沒有 k 圈分割的極圖，根據圈大小的不同，也得到不同邊數的下界。. 中華民國九十四年六月. i.

(3) Extremal Graphs of Ck -decomposition Student: Liang-Chiuan Chen. Advisor: Hung-Lin Fu. Department of Applied Mathematics. Department of Applied Mathematics. National Chiao Tung University. National Chiao Tung University. Hsinchu 300, Taiwan. Hsinchu 300, Taiwan. Abstract A graph G is said to have a Ck -decomposition if E(G) can be partitioned into a collection of subsets each induces a k-cycle. Clearly, if G has a Ck -decomposition, then G is an even graph of order at least k and k divides |E(G)|. The graphs satisfying the above two conditions are called k-sufficient. It is not difficult to see that a k-sufficient graph may not have a Ck -decomposition. In this thesis, at first, we study the k-sufficient r-regular graphs of order n in which Ck -decomposition does not exist. By direct constructions, we show that there are constraints on r with respect to k and n. In order to decompose an arbitrary r-regular graph of 3 n n order n into Ck ’s, r has to be at least 2t+1 4t n, 5 n, 2 , and 2 if k is 2t+1, 4, 2t, and n respectively. On the second part, we also study the extremal k-sufficient graphs which have no Ck -decomposition. As a consequence, ¡n¢ the following results are obtained: (i) If n is even, then ex(n; C3 -decomp.) > 2 − (n − 2) − ²n where ²n = 4 in case that n ≡ 2 or 4 (mod 6) ¡and ¢²n = 5 in case that n ≡ 0 (mod 6). (ii) If n is odd, then ex(n; C3 -decomp.) > n−2 − ²n where ²n = 4 in case that n ≡ 1 2 (mod 6) and ²n = 0 in that n ≡ 3 or 5 (mod 6). (iii) For k > 4, if n is odd, ¡ncase ¢ ex(n; Ck -decomp.) > −2(n−3)−² k,n , where ²k,n ∈ {0, 3, 4, 5, ..., k−1, k+1, k+2}, 2 ¡ ¢ such that n2 ¡−2(n−3)−² is a multiple of k. (iv) For k > 4, if n is even, ex(n; Ck k,n ¢ n n−2 decomp.) > ¡ ¢2 − 2(n − 3) − 2 − ²k,n , where ²k,n ∈ {0, 3, 4, 5, ..., k − 1, k + 1, k + 2}, such that n2 − 2(n − 3) − n−2 2 − ²k,n is a multiple of k.. ii.

(4) Contents Abstract (in Chinese). i. Abstract (in English). ii. Acknowledgement. iii. Contents. iv. List of Figures. v. List of Tables. v. 1 Introduction. 1. 1.1. The Preliminaries in Graph Theory . . . . . . . . . . . . . . . . . . . . . .. 1. 1.2. Cycle Systems and Known Results On Cycle Decomposition . . . . . . . .. 2. 2 Lower Bound of degree r. 6. 3 Lower Bound of ex(n ; Ck -decomp.). 12. 4 Conclusion. 17. iv.

(5) List of Figures 1. G4(Km ) with D = {1}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2. Each edge is Km,m . This graph is G6(Km ) with D = {1, 2}. . . . . . . . . .. 7. 3. 2k-sufficient r-regular graph which has no C2k -decomposition. . . . . . . . 10. 4. ²n = 4 or 5 if n is even. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. 5. ²n = 4 or 0 if n is odd. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. 6. Kn−2 may minus a C²n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. 7. Kn−2 − I minus a C3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. List of Tables 1. The lower bound of r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. 2. The lower bound of ex(n; Ck -decomp.) if n is odd. . . . . . . . . . . . . . . 16. 3. The lower bound of ex(n; Ck -decomp.) if n is even. . . . . . . . . . . . . . . 16. v.

(6) 1. Introduction. Graph decomposition is one of the most important topics in the study of graph theory. The main reason is due to the fact that decomposing the complete graph of order v with multiplicity λ into a collection of complete subgraphs of order k is equivalent to construct a balanced incomplete block design(BIBD), 2-(v, k, λ) design. By replacing the complete subgraphs of order k with k-cycles, we have a λ-fold k-cycle system of order v. Both BIBD and cycle system have been utilized in designing experiments with very high efficiency. Therefore, it is interesting to study the graphs which have a Ck -decomposition and also the graphs which have no Ck -decomposition. We start this thesis with some preliminaries of graph theory.. 1.1. The Preliminaries in Graph Theory. In this section, we first introduce the terminologies and definitions of graphs. For details, the readers may refer to the book ”Introduction to Graph Theory”.[10] A graph G is consisting of a vertex set V (G), an edge set E(G), and a relation that associates with each edge two vertices called its endpoints. A loop is an edge whose endpoints are equal. Multiedges are edges having the same pair of endpoints. A simple graph is a graph without loops or multiedges. In this thesis, all the graphs we consider are simple. The size of the vertex set V (G), |V (G)|, is called the order of G. And the size of the edge set E(G), |E(G)|, is called the size of G. If e = (u, v) is an edge of G, then e is said to be incident to u and v. We also say that u and v are adjacent to each other. For every v ∈ V (G), N (v) denotes the neighborhood of v, that is, all vertices of N (v) are adjacent to v. The degree of v, deg(v) = |N (v)|, is the number of neighborhood of v. We denote that δ(G) is the minimum degree of G and ∆(G) is the maximum degree of G. A cycle is a graph with an equal number of vertices and edges whose vertices can be placed around a circle so that two vertices are adjacent if and only if they appear. 1.

(7) consecutively along the circle. A k-cycle, Ck , is a cycle of size k. A Hamiltonian graph is a graph with a spanning cycle, also called a Hamiltonian cycle which is denoted by Cn where n is the order of the graph. A complete graph is a simple graph whose vertices are pairwise adjacent; the complete graph with n vertices is denoted by Kn . A graph G is bipartite if V (G) is the union of two disjoint independent sets called partite sets of G. A graph G is q-partite if V (G) can be expressed as the union of q independent sets. A complete bipartite graph is a simple bipartite graph such that two vertices are adjacent if and only if they are in different partite sets. When the sets have the sizes s and t, the complete bipartite graph is denoted by Ks,t . If the sets have the same size n, the complete bipartite graph is called balanced, which is denoted by Kn,n . Similarly, the complete q-partite graph is denoted by Ks1 ,s2 ,...,sq and the balanced complete q-partite graph is denoted by Kq(n) where each partite set has n vertices. An even graph is a graph whose degree of vertices are even, and an odd graph is a graph whose degree of vertices are odd. A graph is called r-regular if all its vertices have the same degree r. A subgraph of G is a graph H such that V (H) ⊆ V (G) and E(H) ⊆ E(G). A factor of G is a spanning subgraph of G. A k-factor is a spanning k-regular subgraph. A matching of size k in G is a subgraph of k pairwise disjoint edges. If a matching covers all vertice of G, then it is a perfect matching or 1-factor. A graph G is k-sufficient if G is an even graph of order at least k and the size of G is a multiple of k. A Ck -decomposition of G is a collection of edge-disjoint Ck ’s which partition E(G). A graph G is called Ck -decomposable if G has a Ck -decomposition which is denoted by Ck | G; otherwise, Ck - G.. 1.2. Cycle Systems and Known Results On Cycle Decomposition. If Kn has an m-cycle decomposition, i.e., Cm | Kn , then we refer to this decomposition as an m-cycle system of order n. The study of cycle system dated back to 1847, Kirkman proved the following result.. 2.

(8) Theorem 1.1. [6] A 3-cycle system of the complete graph of order n exists if and only if n ≡ 1 or 3 (mod 6). Since then, the existence of a k-cycle system of order n has attracted quite a few researchers to work on this interesting topic. The following results are well-known now. Theorem 1.2. [9] Let n be an odd integer and m be an even integer with 3 6 m 6 n. The graph Kn can be decomposed into cycles of length m whenever m divides the number of edges in Kn . Theorem 1.3. [1] For positive odd integers m and n with 3 6 m 6 n, the graph Kn can be decomposed into cycles of length m if and only if the number of edges in Kn is a multiple of m. From above three theorems, we can see that the order of complete graph is all odd since the graph must be even. If n is an even integer, then we consider the decomposition of Kn − I where I is a 1-factor of Kn . Theorem 1.4. [9] Let n be an even integer and m be an odd integer with 3 6 m 6 n. The graph Kn − I can be decomposed into cycles of length m whenever m divides the number of edges in Kn − I. Theorem 1.5. [1] For positive even integers m and n with 4 6 m 6 n, the graph Kn − I can be decomposed into cycles of length m if and only if the number of edges in Kn − I is a multiple of m. Therefore, it is interesting to know whether Kn − H can be decomposed into k-cycles where H is a subgraph of Kn such that Kn − H is k-sufficient. The following results deal with the case when H is a 2-regular or 3-regular subgraph. Theorem 1.6. [5] Let F be a 2-regular subgraph of Kn . There exists a C4 -decomposition of Kn − E(F ) if and only if n is odd and 4 divides |E(Kn ) − E(F )|. Theorem 1.7. [2] Let F be a 2-regular subgraph of Kn . There exists a C6 -decomposition of Kn − E(F ) if and only if n is odd and 6 divides |E(Kn ) − E(F )|. 3.

(9) Theorem 1.8. [8] Let U be any 2-factor of Kn , where n is even. Then there exists a 3factor T of Kn with E(U ) ⊂ E(T ) such that Kn − E(T ) admits a hamilton decomposition. If we decompose the other kind of graphs, not necessarily be complete graph, then we have different results. Theorem 1.9. [7] Let F be a set of q vertex-disjoint cycles with the length of the j-th cycle being sj . Then there exists a 2-factor U ∼ = F of Km,m,m , such that Km,m,m − E(U ) Pq has a hamilton decomposition if and only if j=1 sj = 3m. Theorem 1.10. [3] There exists a maximal set S of m edge-disjoint Hamilton cycles in Kn,n if and only if n/4 < m 6 n/2. Theorem 1.11. [4] There exists a maximal set of m hamilton cycles in Kn(p) , if and only if, 1. dn(p − 1)/4e 6 m 6 bn(p − 1)/2c and 2. m > n(p − 1)/4 if (i) n is odd and p ≡ 1 (mod 4), or (ii) p = 2, or (iii) n = 1, except possibly if n = 2m and except possibly if n > 3 is odd, p is odd, and m 6 ((n + 1)(p − 1) − 2)/4. On these results, we can see the degree of these regular graphs are larger than n2 . It seems that if the degree of a graph G is large enough, then we can decompose G into kcycles as long as the graph is k-sufficient. Thus, we are interesting in finding the number r such that an arbitrary k-sufficient r-regular graph which has a Ck -decomposition. For k = 3, the following conjecture by Nash-Williams is worth of mentioning first.. Conjecture(Nash-Williams). Let H be a subgraph of Kn (n 6= 9) such that Kn − H is 3-sufficient and ∆(H) 6 14 (n − 1). Then C3 | Kn − H.. 4.

(10) This conjecture is far from being proved at this moment. But, this upper bound on H or equivalently the lower bound on Kn − H plays an important role in decomposition problems. We shall first focus on the situation when Kn − H or the graph G we consider is r-regular and k-sufficient but G is not able to be decomposed into k-cycles even if G is k-sufficient. Of course, we are looking for r which is as large as possible. In next section, we shall consider the r with respect to the order of G and show that if r is not large enough, then an arbitrary k-sufficient r-regular graph does not have a Ck -decomposition.. 5.

(11) 2. Lower Bound of degree r. Let G be an arbitrary k-sufficient r-regular graph. It is not difficult to realize that to determine whether G can be decomposed into k-cycles or not is not an easy task. Thus, we are interesting in the situation when G is k-sufficient and r-regular but G has no Ck -decomposition. Clearly, we looking for the number “r” as large as possible. First, we introduce a couple of definitions. Definition 2.1. Let G be a graph of order n with V (G) = {v0 , v1 , ..., vn−1 }. Given a bijection function f : V (G) → {0, 1, ..., n − 1} such that f (vi ) = i, 0 6 i 6 n − 1. Define the difference of vi and vj by d(i, j) = min{|j − i|, n − |j − i|}. A graph G of order n is a difference graph G[D] if D ⊆ {1, 2, ..., b n2 c} and E(G) = {(i, i + k). (mod n) | f or all k ∈. D}. Definition 2.2. A graph G is a q-partite-Km graph with a difference set D ⊆ {1, 2, .., b 2q c} if there are q partites G0 , G1 , ..., Gq−1 in G, where each partite Gi , 0 6 i 6 q − 1, is a complete graph of order m. If there are edges between Gi and Gj , 0 6 i < j 6 q − 1, then the edges between Gi and Gj denoted by E(Gi , Gj ) induces a complete bipartite graph S Sq−1 S Km,m . So V (G) = q−1 E(Gi , Gj ) where d(i, j) = {k | i=0 V (Gi ) and E(G) = i=0 E(Gi )∪ ∀ k ∈ D}. The q-partite-Km graph is denoted Gq(Km ) with D ⊆ {1, 2, .., b 2q c}. Moreover, S let E1 (G) = q−1 i=0 E(Gi ) and E2 (G) = E(G) \ E1 (G). Example, let G denote the difference graph G[D]. Then, the graph H1 given by G4(Km ) with D = {1} and the graph H2 given by G6(Km ) with D = {1, 2} are 4-partite-Km graph and 6-partite-Km graph respectively. See Figure 1 and Figure 2 as illustration. Lemma 2.3. If G is a difference graph G[D] of even order n, where D = {i | i is odd}, then G contains no odd cycle. Proof. In a difference graph, a cycle can be formed by two ways. First, the sum of the difference of the edges in cycle is a multiple of the order of G. Second, the sum of the difference of some edges in cycle is equal to the sum of the difference of others. Now, D = {i | i is odd} and n is even. Since the odd sum of odd integers is not an even integer, the proof follows. 6.

(12) 0. 3 m,m m. m. m,m. m,m. m,m m. m. 1. 2. Figure 1: G4(Km ) with D = {1}.. 0. 5 m. m. m. m 1. 4. m. m. 2. 3. Figure 2: Each edge is Km,m . This graph is G6(Km ) with D = {1, 2}.. 7.

(13) From this fact, the following result is easy to see. Corollary 2.4. If G = G2q(Km ) with D = {i | i is odd}, then E2 (G) contains no odd cycle. Lemma 2.5. Consider G = Gq(Km ) and suppose E2 (G) contains no C2k+1 . If 2k × |E1 (G)| < |E2 (G)|, then G is not C2k+1 -decomposable. Proof. Suppose G is C2k+1 -decomposable. Since E2 (G) contains no C2k+1 , we must use at least one edge in E1 (G) and at most 2k edges in E2 (G) to form a C2k+1 . Thus, 2k × |E1 (G)| > |E2 (G)|, a contradiction. Now, we start the constructions with odd cycles decomposition. Proposition 2.6. There is a family of (2k+1)-sufficient r-regular graphs of order n which have no C2k+1 -decomposition, where r =. 2k+1 n 4k. − 1.. Proof. Let G = G4k(Km ) with D = {i | i is odd}, where m = (2k + 1)(2t + 1) for any nonnegative integer t. For example k = 1, G is given by Figure 1 First, we claim that G is (2k+1)-sufficient. Since for all v ∈ V (G), deg(v) = (m − 1) + m × 2k ≡ 0. (mod 2), so G is an even graph. Since |E(G)| = |E1 (G)| + |E2 (G)| =. m(m−1) ×4k +m2 ×k ×4k 2. ≡0. (mod m) ≡ 0. (mod 2k +1), so the size of G is a multiple. of 2k + 1. Next, E2 (G) contains no C2k+1 by Corollary 2.4. And G is not C2k+1 -decomposable since 2k×|E2 (G)| = 2k× m(m−1) ×4k = 4k 2 m(m−1) < 4k 2 m2 = |E2 (G)| (by Lemma 2.5). 2 Hence G is a (2k + 1)-sufficient r-regular graph which has no C2k+1 -decomposition, where r = deg(v) = (2k + 1)m − 1 =. 2k+1 n 4k. − 1.. Corollary 2.7. If every (2k + 1)-sufficient r-regular graphs of order n have C2k+1 decompositions, then r has to be at least. 2k+1 n. 4k. Proof. By the direct construction of Proposition 2.6, there is a family of (2k+1)-sufficient r-regular graphs of order n which have no C2k+1 -decomposition where r =. 2k+1 n 4k. − 1. So. if we want to decompose every (2k + 1)-sufficient r-regular graphs of order n, then r has to be at least. 2k+1 n. 4k. 8.

(14) Besides the construction of Proposition 2.6, there are another two family of graphs which satisfy such conditions. First, let H be a balanced complete bipartite graph of order 4t. Consider G is a S graph of order 4t where V (G) = V (H) and E(G) = E(H) ∪ i E(Ci ) where Ci belongs to partite set. We can choose these Ci properly, such that the minimum degree of G P 2 is as large as possible and G is (2k+1)-sufficient, but i |E(Ci )| < 4t . Then G is not 2k C2k+1 -decomposable by a similar idea of Lemma 2.5. Second, let G be a difference graph G[D] of even order. Choose D = A ∪ B where A = {i | i is odd} and B ⊂ {j | j is even}, but |A| <. |B| . 2k. Then G is not C2k+1 -. decomposable by a similar idea of Lemma 2.5. Proposition 2.8. There is a family of 4-sufficient r-regular graphs of order n which have no C4 -decomposition, where r = 53 n − 1. Proof. Let G = G5(Km ) with D = {1}, where m = 8t + 3 for any nonnegative integer t. Clearly, for all v ∈ V (G), deg(v) = (m − 1) + 2m ≡ 0 m(m−1) 2. × 5 + m2 × 5 = 480t2 + 340t + 60 ≡ 0. (mod 2), and |E(G)| =. (mod 4) . Thus, G is 4-sufficient.. For all i, 0 6 i 6 4, the size of E(Gi , Gi+1 ) is odd, but it is impossible to use an odd number of edges in E(Gi , Gi+1 ) and some edges in E1 (G) to form a C4 . Thus, G is not C4 -decomposable. Hence G is a 4-sufficient r-regular graph which has no C4 -decomposition, where r = deg(v) = 3m − 1 = 53 n − 1. Corollary 2.9. If every 4-sufficient r-regular graphs of order n have C4 -decompositions, then r has to be at least 53 n. For even cycles, the lower bound we obtain is not as good as those we found for odd cycles. Proposition 2.10. There is a family of 2k-sufficient r-regular graphs of order n which have no C2k -decomposition, where r =. n 2. − 1.. 9.

(15) Proof. Let G1 and G2 be the complete graphs of order 4kt + 1 respectively for any nonnegative integer t. Let G be the graph of order 2(4kt+1) where V (G) = V (G1 )∪V (G2 ). Choose E(G) = E(G1 ) ∪ E(G2 ) ∪ {(x1 , y1 ), (x2 , y2 )} \ {(x1 , x2 ), (y1 , y2 )} where x1 , x2 ∈ V (G1 ) and y1 , y2 ∈ V (G2 ). (see Figure 3). x1. y1. x2. y2. G1. G2. Figure 3: 2k-sufficient r-regular graph which has no C2k -decomposition. For all v ∈ V (G), deg(v) = (4kt + 1) − 1 ≡ 0 (mod 2). |E(G)| = (4kt + 1) × 4kt ≡ 0 (mod 2k). So G is 2k-sufficient. Suppose that G is C2k -decomposable, then (x1 , y1 ) and (x2 , y2 ) belong to a C2k in G. Next, we want to use (x1 , y1 ), (x2 , y2 ), q edges in E(G1 ) \ (x1 , x2 ) where 1 6 q 6 2k − 3, and 2k − 2 − q edges in E(G2 ) \ (y1 , y2 ) to form a C2k . Since G is C2k -decomposable and |E(G1 ) \ (x1 , x2 )| = |E(G2 ) \ (y1 , y2 )| =. (4kt+1)×4kt 2. − 1 ≡ −1. (mod 2k) ≡ 2k − 1. (mod 2k), so we must choose q edges properly such that q ≡ 2k − 2 − q ≡ 2k − 1 (mod 2k), it is a contradiction to 1 6 q 6 2k − 3. Thus, G is not C2k -decomposable. Hence G is a 2k-sufficient r-regular graph which has no C2k -decomposition, where r = deg(v) = (4kt + 1) − 1 =. n 2. − 1.. Corollary 2.11. If every 2k-sufficient r-regular graphs of order n have C2k -decompositions, then r has to be at least n2 . 10.

(16) Proof. This follows immediately from Proposition 2.10. Corollary 2.12. If every n-sufficient r-regular graphs of order n have Cn -decompositions, then r has to be at least n2 . Proof. The construction of this proof is the same as Proposition 2.10 except the order of G1 and G2 are 2t+1 for any nonnegative integer t. So, we conclude this section with a table for “r” in which G is an arbitrary r-regular k-sufficient graph but G has no Ck -decomposition.. C3 r. 3 n 4. −1. C4 3 n 5. −1. Table 1: The lower bound of r. C5 C6 ... Ck , k is odd 5 n 8. −1. n 2. −1. .... 11. k n 2(k−1). −1. Ck , k is even n 2. −1. Cn n 2. −1.

(17) 3. Lower Bound of ex(n ; Ck -decomp.). Let F be a given graph. Then we define ex(n; F ) = max{|E(G)| | |V (G)| = n, but G contains no subgraph which induces F}. We call the graph G of order n an extremal graph of F if G contains no subgraph which induces F and |E(G)| = ex(n; F ). In this section, we will study a new topic “extremal graph of Ck -decomposition.” Definition 3.1. We define ex(n; Ck -decomp.) = max{|E(G)| | |V (G)| = n, G is k − suf f icient, but Ck - G}. We call the graph G of order n an extremal graph of Ck decomposition if G satisfies the followings: G is k-sufficient, G is not Ck -decomposable, and |E(G)| = ex(n; Ck -decomp.). In what follows, we obtain the lower bound of ex(n; Ck -decomp.). Lemma 3.2. Let G be a k-sufficient graph. If there is an edge e in G, but e does not lie in any k-cycle in G, then G is not Ck -decomposable. Although the idea of Lemma 3.2 is very simple, it is very useful in proving the following results. Theorem 3.3. If n is even, then ex(n; C3 -decomp.) > ½. ¡ n¢ 2. − (n − 2) − ²n where. ²n = 5 if n ≡ 0 (mod 6) ²n = 4 if n ≡ 2 or 4 (mod 6); and. if n is odd, then ex(n; C3 -decomp.) > ½. ¡n−2¢ 2. − ²n where. ²n = 4 if n ≡ 1 (mod 6) ²n = 0 if n ≡ 3 or 5 (mod 6).. Proof. Let H be the complete graph of order n − 2. Suppose V (H) = {v0 , v1 , ..., vn−3 }, and choose V (H1 ) = {v0 , v1 , ..., v n2 −2 } and V (H2 ) = V (H) \ V (H1 ). If n is even, let G be the graph of order n where V (G) = V (H) ∪ {x, y} and E(G) = E(H) ∪ (x, y) ∪ S S u∈V (H1 ) (x, u) ∪ v∈V (H2 ) (y, v) \ E(C²n ) where E(C²n ) ⊆ E(H). Choose ²n = 5 if n ≡ 0 (mod 6), and ²n = 4 if n ≡ 2 or 4 (mod 6)(see Figure 4). If n is odd, let G be the graph of order n where V (G) = V (H) ∪ {x, y} and E(G) = E(H) ∪ {(x, z1 ), (x, z2 ), (y, z1 ), (y, z2 )} \ E(C²n ) where E(C²n ) ⊆ E(H) and z1 , z2 ∈ V (H). Choose ²n = 4 if n ≡ 1 12. (mod 6), and.

(18) y. x. vn. v 0 v1. vn vn 2. 2. 3 2. v n-3. 4 2. CH n. Figure 4: ²n = 4 or 5 if n is even. ²n = 0 if n ≡ 3 or 5 (mod 6) (see Figure 5). Now, we delete a C²n from Kn−2 to make the graph 3-sufficient.. y. x. v0. v1. v n-4. v n-3. CH n. Figure 5: ²n = 4 or 0 if n is odd.. 13.

(19) Since (x, y) does not lie in any 3-cycle in G, then G is not C3 -decomposable by ¡ ¢ Lemma 3.2. So G is a 3-sufficient graph G with |E(G)| = n2 − 2 × n−2 − ²n = 2 ¡n¢ ¡n−2¢ − (n − 2) − ²n if n is even, and |E(G)| = 2 − ²n if n is odd, but G has no 2 ¡ ¢ C3 -decomposition. Hence ex(n; C3 -decomp.) > n2 −(n−2)−²n if n is even, and ex(n; C3 ¡ ¢ decomp.) > n−2 − ²n if n is odd. 2 ¡ ¢ Theorem 3.4. If n is odd, then ex(n; C4 -decomp.) > n2 − 2(n − 3) − ²n where  ²n = 0 if n ≡ 1 (mod 8)    ²n = 3 if n ≡ 3 (mod 8) ²  n = 6 if n ≡ 5 (mod 8)   ²n = 5 if n ≡ 7 (mod 8); ¡ ¢ if n is even, then ex(n; C4 -decomp.) > n2 − 2(n − 3) − n−2 − 3. 2 Proof.. Let H be the complete graph of order n − 2. If n is odd, let G be the graph of. order n where V (G) = V (H) ∪ {x, y} and E(G) = E(H) ∪ {(x, y), (y, z), (z, x)} \ E(C²n ) where z ∈ V (H), E(C²n ) ⊆ E(H)(see Figure 6). Choose ²n = 0 if n ≡ 1. (mod 8),. ²n = 3 if n ≡ 3. (mod 8).. (mod 8), ²n = 6 if n ≡ 5 (mod 8), and ²n = 5 if n ≡ 7. If n is even, let G be the graph of order n where V (G) = V (H) ∪ {x, y} and E(G) = E(H) ∪ {(x, y), (y, z), (z, x)} \ {F ∪ E(C²n )} where z ∈ V (H), F is a perfect matching of H, and E(C²n ) ⊆ E(H). Choose ²n = 3 if n is even. (see Figure 7) Finally, we delete a C²n to make the graph G 4-sufficient.. y. x. z. CH n. K n2 Figure 6: Kn−2 may minus a C²n .. 14.

(20) y. x. z. C3. K n2 I Figure 7: Kn−2 − I minus a C3 . Since (x, y) does not lie in any 4-cycle in G, then G is not C4 -decomposable by ¡ ¢ Lemma 3.2. So G is a 4-sufficient graph G with |E(G)| = n2 − 2(n − 3) − ²n if n is ¡ ¢ − 3 if n is even,but G has no Ck -decomposition. odd, and |E(G)| = n2 − 2(n − 3) − n−2 2 ¡ ¢ Hence ex(n; C4 -decomp.) > n2 − 2(n − 3) − ²n if n is odd, and ex(n; C4 -decomp.) > ¡n¢ − 2(n − 3) − n−2 − 3 if n is even. 2 2 Similar to the constructions of Theorem 3.4, we can also construct graphs which can not be decomposed into Ck , k > 5. Therefore, a lower bound for ex(n; Ck -decomp.) we have. Theorem 3.5. For k > 5, if n is odd, ex(n; Ck -decomp.) >. ¡n¢ 2. − 2(n − 3) − ²k,n , where. ²k,n ∈ {0, 3, 4, 5, ..., k − 1, k + 1, k + 2}, such that the size of the graph is a multiple of k. If ¡ ¢ n is even, ex(n; Ck -decomp.) > n2 − 2(n − 3) − n−2 − ²k,n , where ²k,n ∈ {0, 3, 4, 5, ..., k − 2 1, k + 1, k + 2}, such that the size of the graph is a multiple of k. Clearly, the above construction also works for the case on Cn -decomposition. Theorem 3.6. For n > 9, if n is odd ex(n; Cn -decomp.) > ¡ ¢ even ex(n; Cn -decomp.) > n2 − 2(n − 3) − n−2 − 7. 2. ¡ n¢ 2. − 2(n − 3) − 6. If n is. To summarize this section, we use the following two tables to depict the study of this 15.

(21) topic.. Table 2: The lower bound of ex(n; Ck -decomp.) if n is odd. C3 ex(n; Ck -decomp.). ¡n¢ 2. Ck , k > 4. − (n + 2) − ²n. ¡ n¢ 2. − 2(n − 3) − ²k,n. Cn ¡ n¢ 2. − 2(n − 3) − 6. Table 3: The lower bound of ex(n; Ck -decomp.) if n is even. C3 ex(n; Ck -decomp.). ¡n−2¢ 2. − ²n. Ck , k > 4 ¡n¢ 2. − 2(n − 3) −. 16. n−2 2. Cn − ²k,n. ¡ n¢ 2. − 2(n − 3) − 7.

(22) 4. Conclusion. From the results obtained in thesis, we have quite a few examples of showing a Ck decomposition is not possible. But, for those graphs, say with large r in degree or with large size, it is not known whether we can decompose them into k-cycles. We shall work on those decompositions in the future. If possible, we would like to prove that the bounds are sharp, especially those bounds on sizes.. 17.

(23) References [1] B. Alspach and H. Gavlas, “Cycle decompositions of Kn and Kn − I,” J. Combin. Theory, Ser. B 81(2001), no.1, 77-99. [2] D. Ashe and C. A. Rodger, “All 2-regular leaves of partial 6-cycle system,” Ars Combinatoria, to appear. [3] D. E. Bryant, S. El-Zanati and C. A. Rodger, “Maximal sets of Hamilton cycles in Kn,n ,” J. Graph Theory 33(2000), no.1, 25-31. [4] M. Daven, J. A. MacDougall and C. A. Rodger, “Maximal sets of Hamilton cycles in complete multipartite graphs,” J. Graph Theory 43(2003), no.1, 49-66. [5] H. L. Fu and C. A. Rodger, “Four-cycle systems with two-regular leaves,” Graphs Combin. 17 (2001), no. 3, 457-461. [6] Rev. T. P. Kirkman, “On the problem in combinations,” Cambr. and Dublin Math. J. 2(1847), 191-204. [7] C. D. Leach and C. A. Rodger, “Hamilton decompositions of complete multipartite graphs with any 2-factor leave,” J. Graph Theory 44(2003), no.3, 208-214. [8] C. D. Leach and C. A. Rodger, “Hamilton decompositions of complete graphs with a 3-factor leave,” Discrete Math 279(2004), no.1-3, 337-344. [9] M. Sajna, “Cycle decomposition III: graphs and fixed length cycles,” J. Combin. Des. 10 (2002), no.1, 27-78. [10] D. B. West, Introduction to Graph Theory 2nd.. 18.

(24)