Burning game

TL;DR

This paper introduces a two-player game modeling fire spread on graphs, analyzes its properties, bounds, and special cases, and explores conjectures and graph products related to the game burning number.

Contribution

It defines the game burning number, establishes bounds and principles, characterizes graphs with small game burning numbers, and investigates related conjectures and graph products.

Findings

Basic bounds on game burning number established

Continuation Principle proved for the game

Graphs with small game burning numbers characterized

Abstract

Motivated by the burning and cooling processes, the burning game is introduced. The game is played on a graph $G$ by the two players (Burner and Staller) that take turns selecting vertices of $G$ to burn; as in the burning process, burning vertices spread fire to unburned neighbors. Burner aims to burn all vertices of $G$ as quickly as possible, while Staller wants the process to last as long as possible. If both players play optimally, then the number of time steps needed to burn the whole graph $G$ is the game burning number $b_g(G)$ if Burner makes the first move, and the Staller-start game burning number $b_g'(G)$ if Staller starts. In this paper, basic bounds on $b_g(G)$ are given and Continuation Principle is established. Graphs with small game burning numbers are characterized and Nordhaus-Gaddum type results are obtained. An analogue of the burning number conjecture for the burning game is considered and graph products are studied.