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Generalized saturation game

Balázs Patkós, Miloš Stojaković, Jelena Stratijev, Máté Vizer

TL;DR

This work initiates and analyzes a two-player game version of the generalized Turán problem, where players alternately claim edges of $K_n$ while preserving $F$-freeness and the final graph is $F$-saturated; the score is the number of copies of a fixed subgraph $H$ present in the final graph. The authors develop sharp asymptotics and bounds for several natural pairs $(F,H)$, including path-star scores, cycle-free ($S_4$) saturation, and $P_5$-saturation, using constructive strategies for Max to maximize and parity- or structure-based arguments for Mini to minimize the score. They introduce and exploit notions such as deficit tracking, path-extension techniques, and TreeBuilder type strategies to obtain both lower and upper bounds, with many results matching up to lower-order terms. The findings illuminate how the combinatorial structure of $F$-saturated graphs constrains $H$-counts in a competitive setting and open avenues for further exploration of more general $(F,H)$ choices and nonfixed targets. The work advances the understanding of interactive extremal problems and their algorithmic implications for competitive graph construction.

Abstract

We study the following game version of the generalized graph Turán problem. For two fixed graphs F and H, two players, Max and Mini, alternately claim unclaimed edges of the complete graph Kn such that the graph G of the claimed edges must remain F-free throughout the game. The game ends when no further edges can be claimed, i.e. when G becomes F-saturated. The H-score of the game is the number of copies of H in G. Max aims to maximize the H-score, while Mini wants to minimize it. The H-score of the game when both players play optimally is denoted by s1(n, #H, F) when Max starts, and by s2(n, #H, F) when Mini starts. We study these values for several natural choices of F and H.

Generalized saturation game

TL;DR

This work initiates and analyzes a two-player game version of the generalized Turán problem, where players alternately claim edges of while preserving -freeness and the final graph is -saturated; the score is the number of copies of a fixed subgraph present in the final graph. The authors develop sharp asymptotics and bounds for several natural pairs , including path-star scores, cycle-free () saturation, and -saturation, using constructive strategies for Max to maximize and parity- or structure-based arguments for Mini to minimize the score. They introduce and exploit notions such as deficit tracking, path-extension techniques, and TreeBuilder type strategies to obtain both lower and upper bounds, with many results matching up to lower-order terms. The findings illuminate how the combinatorial structure of -saturated graphs constrains -counts in a competitive setting and open avenues for further exploration of more general choices and nonfixed targets. The work advances the understanding of interactive extremal problems and their algorithmic implications for competitive graph construction.

Abstract

We study the following game version of the generalized graph Turán problem. For two fixed graphs F and H, two players, Max and Mini, alternately claim unclaimed edges of the complete graph Kn such that the graph G of the claimed edges must remain F-free throughout the game. The game ends when no further edges can be claimed, i.e. when G becomes F-saturated. The H-score of the game is the number of copies of H in G. Max aims to maximize the H-score, while Mini wants to minimize it. The H-score of the game when both players play optimally is denoted by s1(n, #H, F) when Max starts, and by s2(n, #H, F) when Mini starts. We study these values for several natural choices of F and H.
Paper Structure (10 sections, 27 theorems, 7 equations, 1 figure, 2 tables)

This paper contains 10 sections, 27 theorems, 7 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Paths vs stars
  3. $S_4$-saturation games
  4. Counting $P_3$'s
  5. Counting $P_4$'s
  6. Counting $P_5$'s
  7. Counting $P_6$'s
  8. ${\mathcal{C}}$-saturation games
  9. $P_5$-saturation games
  10. Concluding remarks

Key Result

Theorem 1.1

Figures (1)

  • Figure 1: The Minimax game tree, giving an upper bound on the deficit for the three games in Lemma \ref{['l:minimax-tree']}.

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • proof
  • ...and 43 more