The Game of Cycles for Grids and Select Theta Graphs

TL;DR

This paper analyzes the strategic aspects of a cycle-drawing game on graphs, solving previously unknown cases involving stacked polygons and advancing theoretical understanding of the game's dynamics.

Contribution

It introduces new results on winning strategies for complex graph configurations, extending and refining prior theorems and conjectures in the field.

Findings

Solved game strategies for graphs with stacked polygons

Improved upon previous theorems and conjectures

Identified new research directions in the Game of Cycles

Abstract

We are investigating who has the winning strategy in a game in which two players take turns drawing arrows trying to complete cycle cells in a graph. A cycle cell is a cycle with no chords. We examine game boards where the winning strategy was previously unknown. Starting with a $C_{5}$ sharing two consecutive edges with a $C_{7}$ we solve multiple classes of graphs involving "stacked" polygons. We then expand upon and improve previous theorems and conjectures, and offer some new directions of research related to the Game of Cycles. The original game was described by Francis Su in his book Mathematics for Human Flourishing. The first results on the game were published in The Game of Cycles arXiv:arch-ive/04.00776.