TAILIEUCHUNG - Báo cáo toán học: "On a conjecture concerning the Petersen graph"

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: On a conjecture concerning the Petersen graph. | On a conjecture concerning the Petersen graph Donald Nelson Department of Mathematical Sciences Middle Tennessee State University Murfreesboro TN 37132 USA dnelson@ Neil Robertson Department of Mathematics Ohio State University Columbus OH 43210 USA robertso@ Michael D. Plummer Department of Mathematics Vanderbilt University Nashville TN 37240 USA Xiaoya Zhat Department of Mathematical Sciences Middle Tennessee State University Murfreesboro TN 37132 USA xzha@ Submitted Oct 4 2010 Accepted Jan 10 2011 Published Jan 19 2011 Mathematics Subject Classifications 05C38 05C40 05C75 Abstract Robertson has conjectured that the only 3-connected internally 4-con-nected graph of girth 5 in which every odd cycle of length greater than 5 has a chord is the Petersen graph. We prove this conjecture in the special case where the graphs involved are also cubic. Moreover this proof does not require the internal-4-connectivity assumption. An example is then presented to show that the assumption of internal 4-connectivity cannot be dropped as an hypothesis in the original conjecture. We then summarize our results aimed toward the solution of the conjecture in its original form. In particular let G be any 3-connected internally-4-connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord. If C is any girth cycle in G then N C V C cannot be edgeless and if N C V C contains a path of length at least 2 then the conjecture is true. Consequently if the conjecture is false and H is a counterexample then for any girth cycle C in H N C V C induces a nontrivial matching M together with an independent set of vertices. Moreover M can be partitioned into at most two disjoint non-empty sets where we can precisely describe how these sets are attached to cycle C. work supported by NSF Grant DMS-0354554 t work supported by NSA Grant H98230-09-01-0041 THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P20 1 1.

• Báo cáo khoa học
• báo cáo của tạp chí Department of Mathematic
• tài liệu báo cáo nghiên cứu khoa học
• cách trình bày báo cáo
• kiến thức toán học
• báo cáo toán học