Sign games on graphs

TL;DR

This paper introduces the Sign Game played on graphs, analyzing outcomes and strategies for different graph types, contributing to understanding combinatorial game theory on graph structures.

Contribution

It defines the Sign Game on graphs and provides analysis of outcomes and strategies for various graph classes, a novel exploration in graph-based combinatorial games.

Findings

Identified winning strategies for specific graph types

Determined conditions for positive or negative outcomes

Analyzed the impact of graph structure on game results

Abstract

We define the Sign Game as a two-player game played on a simple undirected mathematical graph $G$. The players alternate turns, assigning vertices of $G$ either $1$ or $-1$, and edges take on the value of the product of their endvertices. The game ends when all vertices are assigned values, and the score of the game is the sum of all edge values. One player's goal is to make the score positive while the other's is to make the score negative. In this paper we investigate the game being played on various types of graphs, determining outcomes and winning strategies.