We show Maker wins the Maker-Breaker perfect matching game in n/2 + o(n) turns when the bias is at least n/ln n − f(n)n/(ln n)5/4, for any f going to infinity with n and n sufficiently large (in terms of f).

We show that there is a constant C such that for any b < n/ln n − Cn/(ln n)3/2, Maker can win the Maker-Breaker Hamilton cycle game in n + Cn/√ln n steps.

In 1990, Hendry Conjectured that every Hamiltonian chordal graph is cycle extendable; that is, the vertices of any non-Hamiltonian cycle are contained in a cycle of length one greater. We disprove this conjecture by constructing counterexamples on $n$ vertices for any $n \geq 15$. Furthermore, we show that there exist counterexamples where the ratio of the length of a nonextendable cycle to the total number of vertices can be made arbitrarily small. We then consider cycle extendability in Hamiltonian chordal graphs where certain induced subgraphs are forbidden, notably $P_n$ and the bull.