Burning game
Nina Chiarelli, Vesna Ir\v{s}i\v{c}, Marko Jakovac, William B., Kinnersley, Mirjana Mikala\v{c}ki

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 by the two players (Burner and Staller) that take turns selecting vertices of to burn; as in the burning process, burning vertices spread fire to unburned neighbors. Burner aims to burn all vertices of 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 is the game burning number if Burner makes the first move, and the Staller-start game burning number if Staller starts. In this paper, basic bounds on 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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph theory and applications
