TAILIEUCHUNG - Báo cáo toán học: "On the Spectra of Certain Classes of Room Frames"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: On the Spectra of Certain Classes of Room Frames. | On the Spectra of Certain Classes of Room Frames Jeffrey H. Dinitz Department of Mathematics University of Vermont Burlington Vermont 05405 USA Douglas R. Stinson Computer Science and Engineering Department and Center for Communication and Information Science University of Nebraska-Lincoln Lincoln Nebraska 68588 USA L. Zhu Department of Mathematics Suzhou University Suzhou 215006 People s Republic of China Submitted June 14 1994 Accepted September 14 1994. Abstract In this paper we study the spectra of certain classes of Room frames. The three spectra are incomplete Room squares uniform Room frames and Room frames of type 2ut1. These problems have been studied in numerous papers over the years in this paper we complete the three spectra except for one possible exception in each case. Math reviews classihcation 05B15 1 Introduction Room squares and generalizations have been extensively studied for over 35 years. In 1974 Mullin and Wallis 15 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 a recent survey see 6 . In this paper we study three well-known spectra 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 1 1994 R7 2 Incomplete Room squares This problem asks for which ordered pairs n s does there exist a Room square of side n containing a Room square of side s as a subarray. By considering incomplete Room squares we can allow s 3 or 5 as well. This problem has been under investigation for over 20 years and a history up to 1992 can be found in 6 . Two more recent papers are 19 9 . Uniform Room frames This problem involves the determination of the existence of Room frames of type tu . having u holes of size t . A systematic study of this problem was begun in 1981 in 3 . The history up to 1992 is found in 6 and more recent results appear in 11 1 . Room frames of type 2ut1 Here we are asking for Room frames with one hole of size t and u holes of size 2. This .

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