Counting Cycles with Deepseek

TL;DR

This paper develops a novel AI-assisted method to derive formulas for counting cycles in graphs, combining graph theory and AI prompting strategies, advancing mathematical combinatorics with AI support.

Contribution

It introduces a new approach integrating AI prompting with graph theory to solve the open problem of deriving a computationally efficient cycle count formula.

Findings

AI alone cannot solve the problem without guidance

The method yields new formulas for cycle counting

Step-by-step prompts improve AI problem-solving ability

Abstract

Despite recent progress, AI still struggles on advanced mathematics. We consider a difficult open problem: How to derive a Computationally Efficient Equivalent Form (CEEF) for the cycle count statistic? The CEEF problem does not have known general solutions, and requires delicate combinatorics and tedious calculations. Such a task is hard to accomplish by humans but is an ideal example where AI can be very helpful. We solve the problem by combining a novel approach we propose and the powerful coding skills of AI. Our results use delicate graph theory and contain new formulas for general cases that have not been discovered before. We find that, while AI is unable to solve the problem all by itself, it is able to solve it if we provide it with a clear strategy, a step-by-step guidance and carefully written prompts. For simplicity, we focus our study on DeepSeek-R1 but we also investigate other AI approaches.