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A Graph-Grabbing Game

  • PIOTR MICEK (a1) and BARTOSZ WALCZAK (a1)
Abstract

Two players share a connected graph with non-negative weights on the vertices. They alternately take the vertices (one in each turn) and collect their weights. The rule they have to obey is that the remaining part of the graph must be connected after each move. We conjecture that the first player can get at least half of the weight of any tree with an even number of vertices. We provide a strategy for the first player to get at least 1/4 of an even tree. Moreover, we confirm the conjecture for subdivided stars. The parity condition is necessary: Alice gets nothing on a three-vertex path with all the weight at the middle. We suspect a kind of general parity phenomenon, namely, that the first player can gather a substantial portion of the weight of any ‘simple enough’ graph with an even number of vertices.

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[1]Cibulka J., Kynčl J., Mészáros V., Stolař R. and Valtr P. (2009) Solution to Peter Winkler's pizza problem. In Combinatorial Algorithms (Fiala J., Kratochvíl J. and Miller M., eds), Vol. 5874 of Lecture Notes in Computer Science, Springer, pp. 356367.
[2]Cibulka J., Kynčl J., Mészáros V., Stolař R. and Valtr P. (2010) Graph sharing games: Complexity and connectivity. In Theory and Applications of Models of Computation (Kratochvíl J., Li A., Fiala J. and Kolman P., eds), Vol. 6108 of Lecture Notes in Computer Science, Springer, pp. 340349.
[3]Knauer K., Micek P. and Ueckerdt T. How to eat inline-graphic
$\frac{4}{9}$
of a pizza. Discrete Mathematics, to appear.
[4]Micek P. and Walczak B. Parity in graph sharing games. Submitted.
[5]Rosenfeld M. A gold-grabbing game. Open Problem Garden: http://garden.irmacs.sfu.ca/?q=op/a_gold_grabbing_game.
[6]Winkler P. M. (2008) Problem posed at Building Bridges, a conference in honour of the 60th birthday of László Lovász, Budapest.
[7]Winkler P. M. (2004) Mathematical Puzzles: A Connoisseur's Collection, A. K. Peters.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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