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.

\emph{TxGraffiti} is a data-driven, heuristic-based computer program developed to automate the process of generating conjectures across various mathematical domains. Since its creation in 2017, \emph{TxGraffiti} has contributed to numerous mathematical publications, particularly in graph theory. In this paper, we present the design and core principles of \emph{TxGraffiti}, including its roots in the original \emph{Graffiti} program, which pioneered the automation of mathematical conjecturing. We describe the data collection process, the generation of plausible conjectures, and methods such as the \emph{Dalmatian} heuristic for filtering out redundant or transitive conjectures. Additionally, we highlight its contributions to the mathematical literature and introduce a new web-based interface that allows users to explore conjectures interactively. While we focus on graph theory, the techniques demonstrated extend to other areas of mathematics.

The ability of carefully designed computer programs to generate meaningful mathematical conjectures has been demonstrated since the late 1980s, notably by Fajtlowicz's GRAFFITI program [23]. Indeed, this heuristic-based program was the first artificial intelligence to make significant conjectures in matrices, number theory, and graph theory, attracting the attention of renowned mathematicians like Paul Erdős, Ronald Graham, and Odile Favaron. Inspired by the pioneering work of Fajtlowicz, and by interactions with mathematicians who considered conjectures of GRAFFITI, we developed the TxGraffiti program, a modern conjecturing artificial intelligence named in homage to this rich history of conjectures made by GRAFFITI and now available as an interactive website.