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Tree universality in positional games

Grzegorz Adamski, Sylwia Antoniuk, Małgorzata Bednarska-Bzdęga, Dennis Clemens, Fabian Hamann, Yannick Mogge

TL;DR

This work investigates tree-universality in positional games played on the edge set of the complete graph $K_n$. It develops a tree-embedding framework and proves that, for large $n$, Maker (in Maker-Breaker) and Waiter (in Waiter-Client) can force host graphs containing copies of every tree $T$ on $n$ vertices with maximum degree $\Delta(T) \le \frac{c n}{\log n}$, aligning with random-graph heuristics and improving prior bounds. A central contribution is a tree-universal graph theorem: a graph satisfying a set of structural properties (partition, hub star, robust neighborhoods, and a clique-factor) guarantees universality for all such trees, with explicit constructive strategies for both players to realize these properties. The results extend to related game variants and include new auxiliary results, such as a minimum pair-degree game, highlighting connections between expansion, clique structures, and universal spanning trees. Overall, the paper advances understanding of how robust structural properties enable universal tree-embeddings in adversarial positional games, closely mirroring random graphs in adversarial settings.

Abstract

In this paper we consider positional games where the winning sets are tree universal graphs. Specifically, we show that in the unbiased Maker-Breaker game on the complete graph $K_n$, Maker has a strategy to occupy a graph which contains copies of all spanning trees with maximum degree at most $cn/\log(n)$, for a suitable constant $c$ and $n$ being large enough. We also prove an analogous result for Waiter-Client games. Both of our results show that the building player can play at least as good as suggested by the random graph intuition. Moreover, they improve on a special case of earlier results by Johannsen, Krivelevich, and Samotij as well as Han and Yang for Maker-Breaker games.

Tree universality in positional games

TL;DR

This work investigates tree-universality in positional games played on the edge set of the complete graph . It develops a tree-embedding framework and proves that, for large , Maker (in Maker-Breaker) and Waiter (in Waiter-Client) can force host graphs containing copies of every tree on vertices with maximum degree , aligning with random-graph heuristics and improving prior bounds. A central contribution is a tree-universal graph theorem: a graph satisfying a set of structural properties (partition, hub star, robust neighborhoods, and a clique-factor) guarantees universality for all such trees, with explicit constructive strategies for both players to realize these properties. The results extend to related game variants and include new auxiliary results, such as a minimum pair-degree game, highlighting connections between expansion, clique structures, and universal spanning trees. Overall, the paper advances understanding of how robust structural properties enable universal tree-embeddings in adversarial positional games, closely mirroring random graphs in adversarial settings.

Abstract

In this paper we consider positional games where the winning sets are tree universal graphs. Specifically, we show that in the unbiased Maker-Breaker game on the complete graph , Maker has a strategy to occupy a graph which contains copies of all spanning trees with maximum degree at most , for a suitable constant and being large enough. We also prove an analogous result for Waiter-Client games. Both of our results show that the building player can play at least as good as suggested by the random graph intuition. Moreover, they improve on a special case of earlier results by Johannsen, Krivelevich, and Samotij as well as Han and Yang for Maker-Breaker games.
Paper Structure (17 sections, 19 theorems, 20 equations)

This paper contains 17 sections, 19 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Maker-Breaker games concerning spanning trees
  3. Waiter-Client games concerning spanning trees
  4. Notation
  5. Preliminaries
  6. Probabilistic tools
  7. Maker-Breaker game tools
  8. Waiter-Client game tools
  9. Structural properties of trees
  10. Tree embedding lemmas
  11. A tree universal graph
  12. Maker's strategy
  13. Waiter's strategy
  14. Concluding remarks
  15. Tree universality in Client-Waiter games
  16. ...and 2 more sections

Key Result

Theorem 1.1

There exists a constant $c>0$ such that the following holds for every large enough integer $n$. In the $(1:1)$ Maker-Breaker game on $K_n$, Maker has a strategy to occupy a graph which contains a copy of every tree $T$ with $n$ vertices and maximum degree $\Delta(T)\leq \frac{cn}{\log(n)}$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2: Erdős-Selfridge-Criterion erdos1973combinatorial
  • Lemma 2.3: Mindegree game, Lemma 10 in hefetz2009sharp
  • Lemma 2.4: Degree game, Corollary of Lemma 6 in balogh2009diameter
  • Lemma 2.5: Corollary of Lemma 2.3 in barkey2023multistage
  • Lemma 2.6: $K_5$-factor game, Corollary of Theorem 1.1 in liebenau2022threshold
  • proof
  • Theorem 2.7: Corollary 1.4 in bednarska2013weight
  • ...and 23 more