Resolvable Even-Cycle Systems with a 1-Rotational Automorphism Miwako Mishima, 1 Hung-Lin Fu 2 1 Information and Multimedia Center, Gifu University, 1-1 Yanagido, Gifu 501-1193, Japan, E-mail: miwako@info.gifu-u.ac.jp 2 Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan, Republic of China, E-mail: hlfu@math.nctu.edu.tw Received 30 April 2002; revised 23 June 2003 Abstract: In this article, necessary and sufficient conditions for the existence of a 1- rotationally resolvable even-cycle system of kK v are given, which are eventually for the existence of a resolvable even-cycle system of kK v . # 2003 Wiley Periodicals, Inc. J Combin Designs 11: 394–407, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10058 Keywords: cycle system; extended Skolem sequence; 1-rotationally resolvable 1. INTRODUCTION Let V be the point-set (or the vertex-set) of K v and C a collection of cycles of length m, called m-cycles, whose edges partition the edges of K v . Then the pair ðV ; CÞ is called an m-cycle system of K v or a -fold m-cycle system of order v. Assume ðV ; CÞ to be an m-cycle system of K v . In an automorphism group of ðV ; CÞ, i.e., in a group of permutations on v points leaving the collection C of cycles invariant, if there is an automorphism  of order v  1 with a single fixed point, then the system ðV ; CÞ is said to be 1-rotational. For a 1-rotational m-cycle system of K v , the point-set V can be identified with f1g [ Z v1 , i.e., a fixed point 1 and the residue group of integers modulo v  1. In this case, the automorphism can be represented by  : 1 7! 1; i 7! i þ 1 ðmod ðv  1ÞÞ or  ¼ ð1Þð0; 1; ... ; v  2Þ Correspondence to: Miwako Mishima; E-mail: miwako@info.gifu-u.ac.jp Contract grant sponsor: Ministry of Education, Culture, Sports, Science and Technology (Grant-in-Aid for Young Scientists) (to M.M.); Contract grant numbers: (B) 13780173 and (B) 1570023; Contract grant sponsor: National Science Council of the Republic of China (to H.-L.F.); Contract grant number: NSC-90-2115-M-009-027. # 2003 Wiley Periodicals, Inc. 394