tailieunhanh - Báo cáo toán học: "A Note on Sparse Random Graphs and Cover Graphs"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A Note on Sparse Random Graphs and Cover Graphs. | A Note on Sparse Random Graphs and Cover Graphs Tom Bohman Alan Frieze Miklos Ruszinko Lubos Thoma Department of Mathematical Sciences Carnegie Mellon University Pittsburgh PA 15213 USA tbohman@ alan@ ruszinko@ thoma@ Submitted December 1999 Accepted March 24 2000. Abstract It is shown in this note that with high probability it is enough to destroy all triangles in order to get a cover graph from a random graph Gn p with p K log n n for any constant K 2 3. On the other hand this is not true for somewhat higher densities If p A log n 3 n log log n with A 1 8 then with high probability we need to delete more edges than one from every triangle. Our result has a natural algorithmic interpretation. Keywords random graph cover graph poset Proposed running head Sparse Cover Graphs 1991 Mathematics Subject Classification 05C80 06A07 Supported in part by NSF Grant DMS-9627408. Supported in part by NSF grant CCR-9530974. Permanent Address Computer and Automation Research Institute of the Hungarian Academy of Sciences Budapest 63 Hungary-1518. Supported in part by OTKA Grants T 030059 and T 29074 FKFP 0607 1999. Supported in part by NSF grant DMS-9970622. 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R19 1 Cover Graphs 2 The Hasse diagram of the finite poset P V is the directed graph G V A where u v 2 A iff u - v and there is no z 2 V such that u - z - v. The finite undirected graph G V E is a cover graph iff there exists an orientation of its edges E such that G V E is a diagram of some poset P V . Clearly G V A is the diagram of a poset iff it contains no directed cycles and no directed quasicycles. A directed quasicycle is a cycle with oriented edges in which the reversal of the orientation of a single edge creates a directed cycle. The relationship between cover graphs and graph parameters has been investigated in several papers. B. Descartes 5 as noted in 2 showed that there are cover graphs with