Optimal Strategies for Generalized Saturation Games

Core Concepts

The authors study a game version of the generalized graph Turán problem, where two players, Max and Mini, alternately claim unclaimed edges of the complete graph Kn such that the graph of the claimed edges must remain F-free throughout the game. The H-score of the game is the number of copies of H in the final graph, and the goal is to maximize or minimize this score, respectively.

Abstract

The authors introduce the generalized saturation game, where two players, Max and Mini, alternately claim edges of the complete graph Kn such that the graph of the claimed edges remains F-free throughout the game. The H-score of the game is the number of copies of H in the final graph, and the players aim to maximize or minimize this score, respectively.

The authors first study path-saturation games with star scores. They show that for paths Ps with s ≥ 7 and stars Sℓ with ℓ ≥ 3, the H-score is Θ(nℓ-1). They also determine the exact H-scores for smaller path lengths.

Next, the authors consider S4-saturation games, where they analyze the number of P3's and P5's in the final graph. They provide a complete characterization of the P3-score for n ≤ 12 and give upper and lower bounds for larger n.

The authors then study cycle-free saturation games, where the final graph is a forest. They obtain tight bounds on the Sk-score and P4-score in this setting.

Finally, they address P5-saturation games, determining the number of triangles and paths of length 3 in the final graph.

Stats

n - number of vertices in the complete graph Kn
#H - number of copies of graph H
F - forbidden graph that the game graph must remain free of

Quotes

None.

Key Insights Distilled From

Generalized saturation game

by Balá... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2404.02288.pdf

Deeper Inquiries

How can the techniques and results presented in this paper be extended to other types of forbidden graphs F or target graphs H

The techniques and results presented in the paper can be extended to other types of forbidden graphs F or target graphs H by considering different combinations of F and H. The game variant of the generalized Turán problem can be applied to various graph structures by adjusting the rules and objectives of the game accordingly. For example, different types of cycles, paths, or complete graphs can be used as forbidden graphs F, while various subgraphs or connectivity requirements can be considered as target graphs H. By exploring different combinations of F and H, the game can be adapted to study a wide range of extremal graph theory problems.

What are the computational complexity considerations for determining the optimal strategies in these generalized saturation games

The computational complexity considerations for determining the optimal strategies in these generalized saturation games depend on the specific parameters of the game, such as the size of the graph, the forbidden graph F, and the target graph H. The game-theoretic approach involves analyzing the possible moves of both players to maximize or minimize the score of the game. This analysis may involve exploring different game states, evaluating potential outcomes, and determining the best strategies for each player. The complexity of these calculations can vary based on the size and complexity of the graphs involved, potentially leading to computationally intensive tasks, especially for larger graphs and more complex forbidden and target structures.

Can the game-theoretic approach be applied to other extremal graph theory problems beyond the generalized Turán problem

The game-theoretic approach used in the generalized saturation games can be applied to other extremal graph theory problems beyond the generalized Turán problem. By formulating the problems as combinatorial games with specific rules and objectives, it is possible to analyze various graph properties, such as connectivity, subgraph containment, and edge distribution. This approach can provide insights into optimal strategies for achieving specific graph structures or properties while considering the interactions between players aiming to maximize or minimize certain graph parameters. The game-theoretic perspective offers a unique way to study extremal graph theory problems and can be a valuable tool in exploring different aspects of graph theory.