PHOEG: an online tool for discovery and education in extremal graph theory

Extremal Graph Theory heavily relies on exploring bounds and inequalities between graph invariants, a task complicated by the rapid combinatorial explosion of graphs. Various tools have been developed to assist researchers in navigating this complexity, yet they typically rely on heuristic, probabilistic, or non-exhaustive methods, trading exactness for scalability. PHOEG takes a different stance: rather than approximating, it commits to an exact approach. PHOEG is an interactive online tool (https://phoeg.umons.ac.be) designed to assist researchers and educators in graph theory. Building upon the exact geometrical approach of its predecessor, GraPHedron, PHOEG embeds graphs into a two-dimensional invariant space and computes their convex hull, where facets represent inequalities and vertices correspond to extremal graphs. PHOEG modernizes and expands this approach by offering a comprehensive web interface and API, backed by an extensive database of pairwise non-isomorphic graphs including all graphs up to order 10. Users can intuitively define invariant spaces by selecting a pair of invariants, apply constraints and colorations, visualize resulting convex polytopes, and seamlessly inspect the corresponding drawn graphs. In this paper, we detail the software architecture and new web-based features of PHOEG. Furthermore, we demonstrate its practical value in two primary contexts: in research, by illustrating its ability to quickly identify conjectures or counterexamples to conjectures, and in education, by detailing its integration into university-level coursework to foster student discovery of classical graph theory principles. Finally, this paper serves as a brief survey of the extremal results and conjectures established over the past two decades using this geometric approach.

💡 Research Summary

The paper presents PHOEG, a modern, web‑based platform designed to support both research and teaching in extremal graph theory. PHOEG builds on the exact geometric methodology introduced by its predecessor GraPHedron: every simple, undirected graph of a given order is represented as a point in a two‑dimensional space whose axes correspond to two chosen graph invariants (e.g., chromatic number, clique number). The convex hull of all such points for a fixed order yields a convex polytope; each facet of this polytope encodes a linear inequality between the two invariants, while each vertex corresponds to an extremal graph that attains the bound with equality. By computing the hull exactly, PHOEG provides a complete, minimal set of linear relations for the selected invariant pair, without resorting to heuristics or sampling.

The system stores all pairwise non‑isomorphic graphs up to order 10 (approximately 34 million graphs) in a PostgreSQL database. For each graph, roughly 50 numeric invariants (degree‑based, distance‑based, spectral, etc.) and a dozen Boolean properties have been pre‑computed. Canonical labeling is performed with Nauty to guarantee uniqueness. The backend is a high‑performance RESTful API written in Rust using the Rocket framework; the frontend is a TypeScript/React application styled with Chakra UI, delivering a responsive, accessible user experience.

The user workflow consists of two stages. First, the researcher or student selects the two invariants that define the X‑ and Y‑axes. Optional third‑invariant “coloration” can be added to map a continuous color gradient onto points, a “highlighting” rule can be set to color points that meet a precise invariant value, and arbitrary numeric or Boolean constraints can be applied to filter the dataset. Multiplicity – the number of non‑isomorphic graphs sharing the same coordinate – is visualized as a diamond icon, allowing users to see dense regions versus isolated points.

Second, the platform visualizes the convex polytopes for any chosen orders (2–10) in a left panel. Users can zoom, pan, and synchronize axes across orders for direct comparison. Clicking a point selects it; clicking a facet automatically selects all incident points. The right panel lazily loads the corresponding graph drawings. Eight layout algorithms are available (circular, force‑directed, etc.), and vertices can be colored by degree or other attributes. Additional overlays include invariant annotations, graph signatures, and complement graphs. When coloration or highlighting is active, the same color scheme is applied consistently to both panels, making the relationship between the geometric representation and the concrete graph structures immediately apparent.

The paper demonstrates PHOEG’s research utility through several case studies. Known polyhedral descriptions (e.g., diameter vs. size, Fibonacci index vs. independence number, independence number vs. size for connected graphs) are reproduced, confirming the correctness of the implementation. New invariant pairs are explored: by generating the hull for maximum degree versus average distance, PHOEG quickly suggests conjectural linear bounds; the system then either validates the bound across the entire database or produces counter‑examples, which are displayed instantly. The authors also show how PHOEG can be used to discover previously unknown extremal graphs for custom constraints, illustrating its role as a rapid hypothesis‑testing engine.

In the educational context, PHOEG has been integrated into undergraduate graph theory courses. Students are tasked with defining invariant spaces, applying constraints, and interpreting the resulting polytopes. This hands‑on approach helps them connect abstract theorems (e.g., χ ≥ ω, Turán’s theorem) with concrete graph families, formulate their own conjectures, and test them using the platform. The paper includes sample assignments, student feedback, and evidence that the interactive visualizations improve conceptual understanding and engagement.

Overall, PHOEG delivers an exact, exhaustive exploration of small‑order graphs while providing a user‑friendly, web‑based interface. It bridges the gap between theoretical extremal results and computational experimentation, enabling researchers to generate conjectures, locate counter‑examples, and verify known inequalities with minimal effort. Future work outlined by the authors includes extending the database beyond order 10, adding more invariants (including spectral and chemical descriptors), supporting cloud‑based parallel hull computations for larger orders, and fostering a community‑driven plugin ecosystem for custom analyses.