Juniper Green and the Gallai-Edmonds Decomposition

TL;DR

This paper analyzes the elementary combinatorial game Juniper Green and reveals its connection to the Gallai-Edmonds decomposition of a divisibility graph, uncovering new patterns and characterizations of winning moves.

Contribution

It establishes a novel link between a simple game and the Gallai-Edmonds decomposition, providing new insights into the structure of the divisibility graph.

Findings

Game moves characterized by Gallai-Edmonds decomposition

Discovery of interesting patterns in the decomposition

Connection between elementary game and graph theory

Abstract

Juniper Green is a simple combinatorial game invented by Rob Porteous and popularized by Ian Stewart. It was originally designed to familiarize school children with the concepts of multiplication and division. We analyze this elementary game through a completely different lens and show that it recovers the Gallai-Edmonds decomposition of the divisibility graph on the vertex set $V= \left\{1,2,\dots, n\right\}$. This characterizes the winning moves of the game; as a byproduct, we show that this decomposition seems to have many interesting and curious patterns that are currently unexplained.