Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On Lengths of Rainbow Cycles. | On Lengths of Rainbow Cycles Boris Alexeev Department of Mathematics Massachusetts Institute of Technology Cambridge MA 02139 USA borisa@ Submitted Aug 18 2006 Accepted Oct 27 2006 Published Nov 17 2006 Mathematics Subject Classifications 05C15 05C38 Abstract We prove several results regarding edge-colored complete graphs and rainbow cycles cycles with no color appearing on more than one edge. We settle a question posed by Ball Pultr and Vojtechovsky by showing that if such a coloring does not contain a rainbow cycle of length n where n is odd then it also does not contain a rainbow cycle of length m for all m greater than 2n2. In addition we present two examples which demonstrate that a similar result does not hold for even n. Finally we state several open problems in the area. 1 Introduction A rainbow cycle within an edge-colored graph is a cycle all of whose edges are colored with distinct colors. Rainbow cycles sometimes called colorful or totally multicolored cycles in other sources have been introduced in many different contexts. For example Burr Erdos Sos Frankl and Graham BEGS89 BES 91 studied which graphs may be r-colored for a fixed integer r so that each copy of some subgraph in particular a cycle of a certain length is rainbow. From another perspective Erdos Simonovits and Sós ESS75 investigated a function f n Ck defined as the maximum number of colors in which the edges of the complete graph Kn on n vertices may be colored so that the coloring contains no rainbow k-cycles they conjectured that f n Ck n k 2 2 1 k 1 O 1 for n k 3 and proved the case k 3. Alon Alo83 proved the conjecture for k 4 and derived an upper bound for general k Jiang and West JW03 further improved these bounds and mentioned that the conjecture has been proven for all k 7. Finally Montellano-Ballesteros and Neumann-Lara MBNL05 recently proved the conjecture completely. More research has This research is supported by MIT s Paul E. Gray Endowed Fund for UROP. THE ELECTRONIC .
