Node-Kayles on Trees

TL;DR

This work addresses Grundy-value sequences for Node-Kayles on $n$-regular trees and on graphs formed by grafting rooted trees with a path. It uses the Sprague–Grundy framework and a rooted grafting operation to derive explicit, parity-based formulas and recursive descriptions for the Grundy values, and proves that these sequences are eventually periodic with calculable preperiods and periods. Key contributions include closed-form Grundy values for $\mathcal{G}(T_{\{n\}})$ and $\mathcal{G}(T_{\{n,m\}})$, a parity-based reduction for grafted trees, and concrete periodicity results for various grafting families (e.g., $\mathcal{G}(T_{\{2\}} \stackrel{k}{\cdot - \cdot})$ with preperiod $310$ and period $34$, and $\mathcal{G}(T_{\{2\}} \stackrel{k}{\cdot - \cdot} T_{\{2\}})$ with preperiod $640$ and period $34$). The results extend the catalog of Node-Kayles graph families with known Grundy-periodicity and provide techniques for proving periodicity in octal/graph impartial games.

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.

Figures (6)

• Figure 1: A simple graph $G$
• Figure 2: The induced subgraphs $G_{v_1}, G_{v_2}, G_{v_3}, G_{v_4}, G_{v_5}$ and $G_{v_6}$
• Figure 3: $n$-regular tree $T_{\{2,2,3\}}$
• Figure 4: The rooted grafting graph $G \stackrel{3}{\cdot - \cdot} H$
• Figure 5: Proof of Theorem

Theorems & Definitions (19)

• Theorem 1.1
• Definition 1
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Definition 2
• Theorem 3.1