Báo cáo toán học: "A note on circuit graphs Qing Cui"

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: A note on circuit graphs Qing Cui. | A note on circuit graphs Qing Cui Department of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing 210016 P. R. China cui@ Submitted Oct 12 2009 Accepted Jan 22 2010 Published Jan 31 2010 Mathematics Subject Classifications 05C38 05C40 Abstract We give a short proof of Gao and Richter s theorem that every circuit graph contains a closed walk visiting each vertex once or twice. 1 Introduction We only consider finite graphs without loops or multiple edges. For a graph G we use V G and E G to denote the vertex set and edge set of G respectively. A k-walk in G is a walk passing through every vertex of G at least once and at most k times. A circuit graph G C is a 2-connected plane graph G with outer cycle C such that for each 2-cut S in G every component of G S contains a vertex of C. It is immediate that every 3-connected planar graph G is a circuit graph we may choose C to be any facial cycle of G . In 1994 Gao and Richter 3 proved that every circuit graph contains a closed 2-walk. The existence of such a walk in every 3-connected planar graph was conjectured by Jackson and Wormald 5 . Gao Richter and Yu 4 extended this result by showing that every 3-connected planar graph has a closed 2-walk such that any vertex visited twice is in a vertex cut of size 3. It is easy to see that this also implies Tutte s theorem 7 that every 4-connected planar graph is Hamiltonian. The main objective of this note is to present a short proof of Gao and Richter s result. Theorem 1 Let G C be a circuit graph and let u v G V C . Then there is a closed 2-walk W in G visiting u and v exactly once and traversing every edge of C exactly once. We conclude this section with some notation and terminology. A plane chain of blocks is a graph embedded in the plane with blocks B1 B2 . Bk such that for each i 1 . k 1 Bi and Bi l have a vertex in common no two of which are the same THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 N10 1 and for each j 1 2 . k ui j Bị is in