2. 4-cycle Designs Hung-Lin Fu ( 傅恆霖 ) 國立交通大學應用數學系

3. Motivation The study of graph decomposition has been one of the most important topics in graph theory and also play an important role in the study of the combinatorics of experimental designs (combinatorial designs). Graph theorist can obtain more applications in combinatorial designs than graph decomposition its own.

4. Preliminaries A graph G is an ordered pair (V,E) where V the vertex set is a nonempty set and E the edge set is a collection of subsets of V. In the collection E, a subet (an edge) is allowed to occur many times, such edges are called multi-edges. A complete simple graph on v vertices denoted by K v is the graph (V,E) where E contains all the 2- element subsets of V. Hence, K v has v(v-1)/2 edges. We shall use K v to denote the complete multi- graph with multiplicity, i.e. each edge occurs times.

5. Graph Decomposition H if the edge set of G, E(G), can be partitioned into subsets such that each subset induces a graph in H. For simplicity, we say that G has an H-decomposition. If H =We say a graph G is decomposed into graphs in H if the edge set of G, E(G), can be partitioned into subsets such that each subset induces a graph in H. For simplicity, we say that G has an H-decomposition. If H = {H}, then G has an H-decomposition or an H-design. We shall focus on H a 4- cycle.

6. Balanced Incomplete Block Designs (BIBD) A BIBD or a 2-(v,k, ) design is an ordered pair (X,B) where X is a v- set and B is a collection of k- element subsets (blocks) of X such each pair of elements of X occur together in exactly blocks of B.

7. A Famous Example ! The existence of an STS(v) is equivalent to the existence of a K 3 -decomposition of K v, i.e. decomposing K v into triangles.

8. More General The existence of a 2-(v,k, ) design can be obtained by finding a K k -decomposition of K v. Example: 2K 4 can be decomposed into 4 triangles (1,2,3), (1,2,4), (1,3,4) and (2,3,4). A 2-(4,3,2) design exists and its blocks are: {1,2,3}, {1,2,4}, {1,3,4} and {2,3,4}.

9. Cycle Systems A cycle is a connected 2-regular graph. We use C k to denote a cycle with k vertices and therefore C k has k edges. If G can be decomposed into C k ’s, then we say G has a k-cycle design and denote it by C k | G. If C k | K v, then we say a k-cycle system of order v exists. A 3-cycle system of order v is in fact a Steiner triple system of order v.

10. Necessary conditions If G has a K 3 -decomposition, then the graph must have 3t edges for some t and each vertex is of even degree (even graph). Definition (x-sufficient): A graph G is said to be x-sufficient if x | |E(G)| and G is an even graph. If G has a K 3 -decomposition, then G is 3-sufficient. If G has a decomposition into k-cycles, then G is k-sufficient !

11. Known Results C k | K v if and only if K v is k-sufficient. Let v be even and I is a 1-factor of K v. Then C k | K v – I if and only if K v – I is k-sufficient. After more than 40 years effort, the above two theorems have been proved following the combining results of B. Alspach et al. (2001, JCT(B)

12. 4-cycle Designs A 4-cycle design of the complete graph of order v is also known as a 4-cycle system of order v. Example : v = 9. (0,1,5,3), (1,2,6,4), ….

13. Group Divisible Designs A graph G is a complete m-partite graph if V(G) can be partitioned into m partite sets such that E(G) contains all the edges uv where u and v are from different partite sets. If the partite sets of G are of size n 1, n 2, …, n m, then the graph is denoted by K(n 1,n 2,…,n m ). In case that all partite sets are of the same size n, then we have a balanced complete m-partite graphs denoted by K m(n). A K k -decomposition of K m(n) is a k-GDD and a - fold k-GDD can be defined accordingly. (See it?)

14. A Beautiful 4-cycle Design A 4-cycle design of the complete multipartite graph G exists if and only if G is 4-sufficient. In fact, finding the maximum packing with maximum number of 4-cycles in the complete multipartite graph is also possible. (Billington, Fu, and Rodger, JCD 9)

15. Problem K v – H H For which H K v – H has a 4-cycle decomposition?

16. K v – H H How about this kind of H when |V(H)|  v ?

17. Nash-Williams Conjecture(1970) Let G be a 3-sufficient graph of order n and the minimum degree of G is not less than 3n/4. Then G has a K 3 -decomposition for sufficiently large n. Why 3n/4? (  (H) < n/4 where G = K n – H.)

18. Example: A graph G of order 24m+12 and valency 18m+8. O 6m+3 K 6m+3,6m+3 G c = G can not be decomposed into K 3 ’s.

19. Known Results Theorem(C. Colbourn and A. Rosa, 1986) Let H be a 2-regular subgraph of K v such that v is an odd integer not equal to 9 and v(v-1)/2 - |E(H)| is a multiple of 3. Then K v – H has a K 3 - decomposition. Note: We can also consider the above theorem as packing K v with K 3 ’s such that the leave is H. Let H = C 4  C 5. Then K 9 – H can not be decomposed into K 3 ’s. (See it?)

20. Continued … Theorem(Gustavsson, Ph.D. thesis 1991) Nash-Williams’ conjecture holds for the graphs which are 3-sufficient and minimum degree not less than (1 – 10 -24 )n. Note : I am not able to locate the reference of this result at this moment, the proof is very difficult to check. P.S. 這個問題應該有進展的空間.

21. Problems Let v be an even integer and H be an odd spanning forest of K v such that K v – H is 3-sufficient. Then K v – H has a K 3 - decomposition. ( 我最想解決的問題.) Let v be an even integer and H be an odd spanning subgraph of K v such that  (H) is at most 3 and K v – H is 3-sufficient. Then K v – H has a K 3 -decomposition.

22. More 4-cycle Designs Let H be a 2-regular subgraph of K v where v is odd. Then K v – H has a C 4 -decomposition if and only if v(v-1)/2 - |E(H)| is a multiple of 4 (K v – H is 4-sufficient). (Fu and Rodger, GC 2001) Surprisingly: If H is a spanning forest of K v where v is even, then K v – H has a C 4 - decomposition iff K v – H is 4-sufficient. (Fu and Rodger, JGT 2000)

23. Continued … Let H be an odd graph with  (H) not greater than 3. Then K v – H has a C 4 -decomposition if and only if K v – H is 4-sufficient except two special cases when v = 8. (C.M. Fu, Fu, Rodger and Smith, DM 2004) Conjecture(Fu) Let H be a subgraph of K v with  (H) < v/4 and 3  k  v. Then K v – H has a C k - decomposition if and only if K v – H is k- sufficient.  Why v/4?

24. An example for k = 4 K 8 – H can not be decomposed into 4-cycles. H :

25. Another Evidence Let H be a 2-regular subgraph of K v. Then K v – H has a C 6 -decomposition if and only if K v – H is 6-sufficient. (Ashe, Fu and Rodger, Ars Combin.) Let H be a spanning odd forest of K v where v is even. Then K v – H has a C 6 - decomposition if and only if K v – H is 6- sufficient. (Ashe, Fu and Rodger, DM 2004)

26. Pentagon Designs Compare to 4-cycle systems or 3-cycle systems, the study of 5-cycle systems is harder. It takes a long while to find the necessary and sufficient conditions to decompose a complete 3- partite graph into C 5 ’s. (Billington et al.) Problem: Let H be a 2-regular subgraph of K v such that v is and odd integer, v  5 and v(v-1)/2 - |E(H)| is a multiple of 5. Then K v – H has a C 5 - decomposition. (K v – H is 5-sufficient.)

27. How many 4-cycles in a graph? If we have a 4-cycle design of G, then we have a bunch of 4-cycles in G. Even if G does not have a 4-cycle design, G many have some 4-cycles in there. So, it is interesting to know for which G with size e(G) as large as possible and G contains no 4-cycles !

28. Extremal Graphs Fixed the number of vertices “n”. A graph G of order n with maximum number of edges such that G contains no subgraph H is an extremal graph with forbidden graph H. If H is 3-cycle, then the extremal graph is the “balanced” complete bipartite graph of order n. If H is the complete graph of order k+1, then the extremal graph is the “balanced” complete k- partite graph of order n. For example, if n = 20 and k is 6, then the graph is K(4,4,3,3,3,3).

29. Tougher Case : 4-cycle Let ex(n,H) denote the size of an extremal graph with forbidden graph H. If H is a 3-cycle, then ex(n,H) is about n 2 /4. If H = K r,r, then ex(n,H) is about cn 2 – 1/r where c is a suitable constant (who knows?) Now, how about r = 2?

30. Partial Results Let q be a prime power. Then ex(n,C 4 ) is equal to n(1 + (4n-3) 1/2 )/4 where n = q 2 + q + 1. (From the existence of a projective plane of order q.) How about the cases n is not of this form?

31. Zarankiewicz Problem Let G 2 (m,n) denote the set of all bipartite graphs with partite sets A and B such that |A| = m and |B| = n. Let z(m,n;s,t) denote the maximum size of graphs in G 2 (m,n) which contains no subgraph K s,t. Can we solve the case when s = t = 2?

32. C 4 – Saturated Graphs A graph G is C 4 – saturated if any proper supergraph defined on V(G) contains a 4-cycle, i.e. adding any edge to G creates a 4-cycle. C 5 is C 4 – saturated. K m,n contains a C 4 – saturated subgraph of size m+n-1. We need a C 4 – saturated graph of maximum size!

33. Partial 2-Designs A partial 2-design is a pair (X,B) where X is a nonempty finite set and B is a collection of subsets(blocks) of X such that each pair of two distinct elements occur together in at most one block in B. Let m  n. Then a C 4 – saturated subgraph of K m,n corresponds to a partial 2-design. (?)

34. z(n,n;2,2) Theorem(Reiman,1958) z(n,n;2,2) is not greater than n[1+(4n-3) 1/2 ]/2 and the equality holds if n = q 2 + q + 1 where q is a prime power. The projective plane of order q corresponds to an extremal graph. See it? Problem Is this the only situation that we have the equality? (n)

35. z(m,n;2,2) Theorem (Bryant and Fu) Let (X,B) be a partial 2-design such that |X| = n and B = {B 1, B 2,..., B m }. Then the corresponding graph of (X,B) is a C 4 – saturated subgraph of K m,n with maximum size if the blocks in B contain either k or k+1 elements and

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