tailieunhanh - Báo cáo toán học: "How long can a graph be kept planar"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: How long can a graph be kept planar? | How long can a graph be kept planar V. Anuradha Chinmay Jain Jack Snoeyink Tibor Szabo IIT Bombay IIT Bombay UNC Chapel Hill ETH Zurich Submitted Nov 2 2007 Accepted Apr 25 2008 Published May 5 2008 Mathematics Subject Classification 91A24 Abstract The graph non- planarity game is played on the complete graph Kn between an Enforcer and an Avoider each of whom take one edge per round. The game ends when the edges chosen by Avoider form a non-planar subgraph. We show that Avoider can play for 3n 26 turns improving the previous bound of 3n 28pn. 1 Introduction The non- planarity game is a positional game between two players Avoider and Enforcer played on the edges of the complete graph on n vertices. The game consists of a sequence of rounds in each round Enforcer occupies one previously unoccupied edge of Kn followed by Avoider occupying one. For clarity let us assume that Enforcer starts the game but this is not crucial our result applies even if Avoider does. Avoider seeks to avoid creating a non-planar graph - he aims that his chosen edges should have a planar embedding. Enforcer chooses edges that makes Avoider s task more difficult - she tries to force Avoider s graph to become non-planar. Independent of her strategy Enforcer inevitably succeeds by round 3n 5 when Avoider s graph has simply too many edges to be planar. Jozsef Balogh 1 gave a simple strategy for Enforcer that makes Avoider fail already by turn 3n 7. Once Avoider choses his first edge ab Enforcer pairs va vb for all vertices v a b and whenever Avoider chooses one edge of a pair Enforcer chooses the other. This ensures that ab cannot participate in any triangle so contracting ab removes a single edge from Avoider s graph and leaves the result planar. Thus Avoider has at most 3 n 1 6 1 3n 8 edges. From the other side Avoider can fix an arbitrary triangulation on the n vertices and make sure that he selects at least half of its edges during his turns thus delaying his defeat for at least 3 n 6 .