On the Hamiltonicity of the $k$-regular graph game

TL;DR

This paper investigates a graph game where two players alternately add edges without exceeding degree $k$, revealing that for $k=3$ players can avoid Hamilton cycles, while for $k extgreater 3$ they can force Hamiltonicity.

Contribution

It establishes the threshold at which players can control the Hamiltonicity of the resulting graph based on the degree limit $k$, providing new insights into graph game strategies.

Findings

For $k=3$, players can avoid Hamilton cycles.

For $k extgreater 3$, players can force Hamiltonian graphs.

The degree constraint $k$ determines the game's outcome.

Abstract

We consider a game played on an initially empty graph where two players alternate drawing an edge between vertices subject to the condition that no degree can exceed $k$. We show that for $k=3$, either player can avoid a Hamilton cycle, and for $k\geq4$, either player can force the resulting graph to be Hamiltonian.