Jiuqiang Liu

6-regular Cayley graphs on abelian groups of odd order are hamiltonian decomposable

Alspach conjectured that any 2k-regular connected Cayley graph on a finite abelian group A has a hamiltonian decomposition. In this paper, the conjecture is shown true if k=3, and the order of A is odd.

Pseudo-Cartesian product and Hamiltonian decompositions of Cayley graphs on Abelian groups

Abstract Alspach has conjectured that any 2 k -regular connected Cayley graph cay ( A , S ) on a finite abelian group A can be decomposed into k hamiltonian cycles. In this paper we generalize a result by Kotzig that the cartesian product of any two cycles can be decomposed into two hamiltonian cycles and show that any pseudo-cartesian product of two cycles can be decomposed into two hamiltonian cycles. By applying that result we first give an alternative proof for the main result in (Bermond et al., 1989), including the missing cases, and then we show that the conjecture is true for most 6-regular connected Cayley graphs on abelian groups of odd order and for some 6-regular connected Cayley graphs on abelian groups of even order.

On bandwidth and edgesum for the composition of two graphs

Abstract The composition of two graphs G and H, written G[H], is the graph with vertex set V(G) × V(H) and with (u1, v1) adjacent to (u2, v2) if either u1 is adjacent to u2 in G or u1 = u2 and v1 is adjacent to v2 in H. In this paper, we investigate the bandwidth problem for the composition of two graphs and obtain the bandwidth for several classes of graphs of the forms Kn[G] and K1,n[G]. We also provide optimal numberings which solve the graph edgesum problem for Kn[Pm] and Kn[Cm].

Set systems with cross L-intersection and k-wise L-intersecting families

We prove some results involving cross L-intersections of two families of subsets of [n]={1,2,...,n}. As a consequence, we derive the following results: (1) Let L={l1,l2,...,ls} be a set of s positive integers. If F={F1,F2,...,Fm} is a family of subsets of X=[n] satisfying |Fi-Fj|@?L for i j, then [emailxa0protected][emailxa0protected]?i=0sn-1i. (2) Let p be a prime, k>=2, and L={l1,l2,...,ls} and K={k1,k2,...,kr} be two disjoint subsets of {0,1,...,p-1}. Suppose F is a family of subsets of [n] such that |Fi|(modp)@?K for all F[emailxa0protected]?F and |F[emailxa0protected][emailxa0protected]?Fk|(modp)@?L for any collection of k distinct sets from F. If n>(r+1)(s-2r+2), then |F|@?(k-1)@?i=s-2r+1sn-1i. The first result improves a result of Frankl about families with given difference sizes between subsets and the second result gives an improvement to a theorem by Grolmusz-Sudakov and a theorem by W. Cao, K.W. Hwang, and D.B. West.

Survey of Double Vertex Graphs

Abstract.u2002Let G be a (V,E) graph of order p≥2. The double vertex graphU2(G) is the graph whose vertex set consists of all 2-subsets of V such that two distinct vertices {x,y} and {u,v} are adjacent if and only if |{x,y}∩{u,v}|=1 and if x=u, then y and v are adjacent in G. For this class of graphs we discuss the regularity, eulerian, hamiltonian, and bipartite properties of these graphs. A generalization of this concept is n-tuple vertex graphs, defined in a manner similar to double vertex graphs. We also review several recent results for n-tuple vertex graphs.

Cross L-intersecting families on set systems

Let L={l1,l2,...,ls} be a set of s positive integers. Suppose that A={A1,A2,...,Am} and B={B1,B2,...,Bm} are two collections of subsets of [n]={1,2,...,n} such that |Ai@?Bj|@?L whenever i j. If the set systems satisfy one of the following conditions: (1) |Ai@?Bi|@?L implies Ai=Bi (2) |Ai@?Bj|@?|Aj@?Bj| with equality possible only when |Ai@?Bi|>|Aj@?Bj| for i j, then we bound m as m@?n-1s+n-1s-1+...+n-10. This result extends Snevilys theorem to cross L-intersecting two families.

Set systems with positive intersection sizes

Abstract In this paper, we derive a best possible k -wise extension to the well-known Snevily theorem on set systems (Snevily, 2003) which strengthens the well-known theorem by Furedi and Sudakov (2004). We also provide a conjecture which gives a common generalization to all existing non-modular L -intersection theorems.

Families of vector spaces with r-wise L-intersections

Abstract In this paper, we extend the well-known Frankl–Ray-Chaudhuri–Wilson Theorem on vector spaces to r -wise L -intersecting families, in both modular version and non-modular version. We also provide an upper bound on L -intersecting families of vector spaces which improves the Frankl–Ray-Chaudhuri–Wilson Theorem.

Preface: Volume 11

Abstract The Ninth Quadrennial International Conference on Graph Theory, Combinatorics, Algorithms and Applications was held June 4-9, 2000 at Western Michigan University. This conference has been held every four years since 1968, hosted by the Department of Mathematics and Statistics at Western Michigan University. The 1996 Conference attracted nearly 300 participants from all over the world (including Canada, China, Denmark, England, France, Hungary, India, Israel, Italy, Japan, Korea, Germany, the Netherlands, Puerto Rico, Saudi Arabia, Scotland, South Africa, and Spain) and included five days of principal, invited and contributed presentations by approximately 200 speakers. Acknowledgments The excellent support of Western Michigan University, Dr. Elson S. Floyd, President Dr. Diether H. Haenicke, Past President (with his generous support of the Sixth, Seventh, Eighth, and the Ninth International Conferences) The continued support of the Division of Research and Sponsored Programs, Dr. Donald E. Thompson, Vice President The fine support of the College of Arts and Sciences, Dr. Elise Jorgens, Dean The overall support of the Department of Mathematics and Statistics, Dr. John Petro, past Chair Dr. Jay Wood, Chair The fine financial support of the Number Theory Foundation, Dr. Carl Pomerance, Director Ms. Ethel Rathburn, Secretary The special assistance of our friends and co-directors, Dr. Ronald L. Graham, Dr. John Petro and Dr. Allen J. Schwenk and associate directors Jiuqiang Liu, Michael Raines, and Ping Zhang. The support and assistance of our graph theory colleagues, Professors Arthur White and Gary Chartrand. The fine assistance of colleagues around the world who assisted with refereeing of the manuscripts. The special administrative assistance of Cheryl Peters and the fine secretarial assistance of Maryann Bovo and Margo Chapman. The dedicated work of our conference assistants, Vince Castellana, Linda Eroh, Raluca Muntean and Wendy Weaver. The editors apologize in advance for any oversight in the acknowledgments and any errors in the manuscripts. Y.A. D.R.L. J.L. D.J. Dedication For so many years and after many conferences, when leaving Kalamazoo, boarding the airplane, Paul Erdos would say “Maybe you can hold the next Conference in my memory.” Upon his passing shortly after the 8th Kalamazoo Conference, the decision to proceed with a 9th Conference in memory of Paul Erdos was easy and direct. Thus the Erdos 2000 Conference was held during June 4-9, 2000 in Kalamazoo. So it is with special devotion and honor that we dedicate these Proceedings to him. Another special friend and teacher of many colleagues here was the late Edward A. Nordhaus, to whom these special proceedings are also dedicated.