Đang chuẩn bị nút TẢI XUỐNG, xin hãy chờ
Tải xuống
Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A note on the non-colorability threshold of a random graph. | A note on the non-colorability threshold of a random graph Alexis C. Kaporis Lefteris M. Kirousis and Yannis C. Stamatiou University of Patras Department of Computer Engineering and Informatics Rio 265 00 Patras Greece. e-mail kaporis kirousis stamatiu @ceid.upatras.gr Third author also Computer Technology Institute 61 Riga Feraiou Str. GR 262 21 Patras Greece. Submitted April 6 2000 Accepted May 23 2000 Abstract In this paper we consider the problem of establishing a value r0 such that almost all random graphs with n vertices and rn edges r r0 are asymptotically not 3-colorable. In our approach we combine the concept of rigid legal colorings introduced by Achlioptas and Molloy with the occupancy problem for random allocations of balls into bins. Using the sharp estimates obtained by Kamath et al. of the probability that no bin is empty after the random placement of the balls and exploiting the relationship between the placement of balls and the rigid legal colorings we improve the value r0 2.522 previously obtained by Achlioptas and Molloy to ro 2.495. 1 Introduction In this paper we consider the problem of computing the smallest value r0 such that almost all graphs with rn edges r r0 are not 3-colorable. We say that a graph G V E is 3-colorable if the set V of its vertices can be partitioned into 3 nonempty cells V1 V2 and V3 such that no two vertices of the same partition are adjacent. This partition is called a 3-coloring of G and the vertices of the set Vj j 1 2 3 are said to have color j. 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R29 2 Like many other combinatorial problems on random structures e.g. formulas graphs etc. there appears that the property of a graph being 3-colorable exhibits a threshold behavior. That is it is believed and supported by experimental evidence that there exists a critical constant rc such that a randomly generated graph with n vertices and rc e n edges is 3-colorable with high probability while the opposite is true for a .
