Analysis of Odd/odd vertex removal games on special graphs

TL;DR

This paper studies the Odd/odd vertex removal game on special graphs, proving a parity-based Grundy value characterization for bipartite graphs and confirming the existence of graphs for all Grundy values.

Contribution

It generalizes a conjecture by Shelton by showing bipartite graphs have Grundy values 0 or 1 based on edge parity and confirms graphs exist for every Grundy value.

Findings

Bipartite graphs have Grundy value 0 or 1 depending on edge parity

Confirmed the existence of graphs for every Grundy value

Extended previous conjectures on vertex removal games

Abstract

We analyze the Odd/odd vertex removal game introduced by P. Ottaway. We prove that every bipartite graph has Grundy value 0 or 1 only depending on the parity of the number of edges in the graph, which is a generalization of a conjecture of K. Shelton. We also answer a question originally posed by both Shelton and Ottaway about the existance of graphs for every Grundy value. We prove that this is indeed the case.