In Reverie Together: Ten Years of Mathematical Discovery with a Machine Collaborator

Abstract

We present four open conjectures in graph theory generated by the automated conjecturing system \texttt{TxGraffiti}. Each conjecture is concise, grounded in natural graph invariants, and empirically validated across hundreds of graphs. Despite extensive effort, these statements remain unresolved--defying both proof and counterexample. They are not only mathematical challenges but creative expressions--born of symbolic pattern recognition and mathematician-defined heuristics, refined through years of human dialogue, and now offered back to the community as collaborative artifacts. These conjectures invite not only formal proof, but also reflection on how machines can evoke wonder, spark curiosity, and contribute to the raw material of discovery. By highlighting these problems, we aim to inspire both human mathematicians and AI systems to engage with them--not only to solve them, but to reflect on what it means when machines participate meaningfully in the creative process of mathematical thought.