Node-Kayles on Trees

TL;DR

This paper studies the Grundy value sequences of Node-Kayles on n-regular trees and related graphs, deriving formulas, recursive relations, and proving eventual periodicity of these sequences.

Contribution

It provides explicit formulas, recursive relations, and proves the eventual periodicity of Grundy value sequences for specific classes of graphs in Node-Kayles.

Findings

Grundy sequences are eventually periodic.

Explicit formulas for Grundy values of n-regular trees.

Recursive relations for Grundy sequences.

Abstract

Node-Kayles is a well-known impartial combinatorial game played on graphs, where players alternately select a vertex and remove it along with its neighbors. By the Sprague-Grundy theorem, every position of an impartial game corresponds to a non-negative integer called its Grundy value. In this paper, we investigate the Grundy value sequences of $n$-regular trees as well as graphs formed by joining two $n$-regular trees with a path of length $k$. We derive explicit formulas and recursive relations for the associated Grundy value sequences. Furthermore, we prove that these sequences are eventually periodic and determine both their preperiod lengths and their periods.