Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: The Spectra of Certain Classes of Room Frames: The Last Cases. | The Spectra of Certain Classes of Room Frames The Last Cases Jeffrey H. Dinitz and Gregory S. Warrington Department of Mathematics and Statistics University of Vermont Burlington Vermont . 05405 Submitted Nov 13 2009 Accepted May 5 2010 Published May 20 2010 Mathematics Subject Classification 05B15 Abstract In this paper we study the spectra of certain classes of Room frames. The three spectra which we study are incomplete Room squares uniform Room frames and Room frames of type 2 7. b These problems have been studied in numerous papers over the years in this paper we complete the three spectra. In addition we find a Howell cube of type H3 6 10 . This corrects a previous claim of nonexistence of this design. 1 Introduction Room squares and generalizations have been extensively studied for over 40 years. In 1974 Mullin and Wallis 12 showed that the spectrum of Room squares consists of all odd positive integers other than 3 or 5 however many other related questions have remained unsolved. For an extensive survey from 1992 of Room squares and related designs we refer the reader to 5 . In the 1994 paper by Dinitz Stinson and Zhu 6 the authors studied three well-known spectra of designs closely related to Room squares and in each instance left exactly one unsolved case. In this paper we will prove the existence of each of these designs. Also in the 1986 paper by A. Rosa and D. Stinson 13 it was claimed that there is no Howell cube on any 6-regular graph on 10 points. In Section 2 we disprove this claim by exhibiting such a cube. We begin with the definitions. Let S be a set and let Si . Sn be a partition of S. An S1 . Sn -Room frame is an S X S array F indexed by S which satisfies the following properties THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R74 1 1. every cell of F either is empty or contains an unordered pair of symbols of S 2. the subarrays Si X Si are empty for 1 i n these subarrays are referred to as holes 3. each symbol x G Si occurs once in row or
