The Topology of Competitively Constructed Graphs

  • Alan Frieze
  • Wesley Pegden
Keywords: Graph theory, Combinatorial games, Planar graphs, Graph minors


We consider a simple game, the $k$-regular graph game, in which players take turns adding edges to an initially empty graph subject to the constraint that the degrees of vertices cannot exceed $k$. We show a sharp topological threshold for this game: for the case $k=3$ a player can ensure the resulting graph is planar, while for the case $k=4$, a player can force the appearance of arbitrarily large clique minors.
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