# A Systematization of the Wagner Framework: Graph Theory Conjectures and Reinforcement Learning

arXiv: https://arxiv.org/abs/2406.12667 · HTML: https://mlsanity.com/paper/2406.12667/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in RL for games and combinatorial optimization, one of 15 directions in Deep reinforcement learning methods.
- Builds on 8 of its 19 references that are in the map.
- First posted on arXiv 18 Jun 2024.

## Placed in

- Deep reinforcement learning methods / [RL for games and combinatorial optimization](https://mlsanity.com/deeprl/rl-for-games-and-combinatorial-optimization/)

## Builds on (most cited first, 8 of 8)

- [OpenAI Gym](https://mlsanity.com/paper/1606.01540/) (2016)
- [Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm](https://mlsanity.com/paper/1712.01815/) (2017)
- [Constructions in combinatorics via neural networks](https://mlsanity.com/paper/2104.14516/) (2021)
- [Finding Increasingly Large Extremal Graphs with AlphaZero and Tabu Search](https://mlsanity.com/paper/2311.03583/) (2023)
- [SAI, a Sensible Artificial Intelligence that plays Go](https://mlsanity.com/paper/1809.03928/) (2018)
- [Increasing biases can be more efficient than increasing weights](https://mlsanity.com/paper/2301.00924/) (2023)
- [Score vs. Winrate in Score-Based Games: which Reward for Reinforcement Learning?](https://mlsanity.com/paper/2201.13176/) (2022)
- [GloNets: Globally Connected Neural Networks](https://mlsanity.com/paper/2311.15947/) (2023)

## Built on by (most cited first, 1 of 1)

- [RLGT: A reinforcement learning framework for extremal graph theory](https://mlsanity.com/paper/2602.17276/) (2026)

## Abstract (arXiv)

In 2021, Adam Zsolt Wagner proposed an approach to disprove conjectures in graph theory using Reinforcement Learning (RL). Wagner's idea can be framed as follows: consider a conjecture, such as a certain quantity f(G) < 0 for every graph G; one can then play a single-player graph-building game, where at each turn the player decides whether to add an edge or not. The game ends when all edges have been considered, resulting in a certain graph G_T, and f(G_T) is the final score of the game; RL is then used to maximize this score. This brilliant idea is as simple as innovative, and it lends itself to systematic generalization. Several different single-player graph-building games can be employed, along with various RL algorithms. Moreover, RL maximizes the cumulative reward, allowing for step-by-step rewards instead of a single final score, provided the final cumulative reward represents the quantity of interest f(G_T). In this paper, we discuss these and various other choices that can be significant in Wagner's framework. As a contribution to this systematization, we present four distinct single-player graph-building games. Each game employs both a step-by-step reward system and a single final score. We also propose a principled approach to select the most suitable neural network architecture for any given conjecture, and introduce a new dataset of graphs labeled with their Laplacian spectra. Furthermore, we provide a counterexample for a conjecture regarding the sum of the matching number and the spectral radius, which is simpler than the example provided in Wagner's original paper. The games have been implemented as environments in the Gymnasium framework, and along with the dataset, are available as open-source supplementary materials.