# On the Odd Cycle Game and Connected Rules

## TL;DR

This paper analyzes a positional game where Maker aims to claim all edges of an odd cycle in a complete graph, establishing new bounds for Maker's winning strategy and exploring the game under connected rules and different variants.

## Contribution

It improves the known bounds for Maker winning the odd cycle game and introduces connected rules, extending analysis to multiple game variants.

## Key findings

- Maker wins if b ≤ ((4 - √6)/5 + o(1)) n
- Introduction of connected rules in the odd cycle game
- Analysis of game under Maker-Breaker and Client-Waiter variants

## Abstract

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.