The Topology of Competitively Constructed Graphs

Alan Frieze, Wesley Pegden
2014 Electronic Journal of Combinatorics  
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.
doi:10.37236/3942 fatcat:dtvtjc2bnzdjfldlqifzczb7ey