On the Hamiltonicity of the $k$-regular graph game
  We consider a game played on an initially empty graph where two players alternate drawing an edge between vertices subject to the condition that no degree can exceed $k$. We show that for $k=3$, either player can avoid a Hamilton cycle, and for $k\geq4$, either player can force the resulting graph to be Hamiltonian. 
Jeremy Meza and Samuel Simon
math.CO