  We study the positional game where two players, Maker and Breaker,
alternately select respectively $1$ and $b$ previously unclaimed edges of
$K_n$. Maker wins if she succeeds in claiming all edges of some odd cycle in
$K_n$ and Breaker wins otherwise. Improving on a result of Bednarska and
Pikhurko, we show that Maker wins the odd cycle game if $b \leq ((4 -
\sqrt{6})/5 + o(1)) n$. We furthermore introduce "connected rules" and study
the odd cycle game under them, both in the Maker-Breaker as well as in the
Client-Waiter variant.

Combinatorics