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

2026-03-28

AI Summary

This work addresses the challenge in extremal graph theory that the explosive growth in the number of graphs with increasing order hinders systematic exploration of inequalities and extremal structures. The authors propose an exact geometric approach: embedding all non-isomorphic graphs of order up to ten into a two-dimensional invariant space and automatically uncovering extremal graphs and valid inequalities via convex hull computation. They construct a complete database of such non-isomorphic graphs and provide an interactive web platform with API support, enabling conjecture verification and pedagogical applications. Requiring no heuristic assumptions, this method successfully reproduces and extends several classical results and has been integrated into university curricula, significantly enhancing both research efficiency and teaching effectiveness in extremal graph theory.

Abstract

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.

Problem

• Extremal Graph Theory
• graph invariants
• combinatorial explosion
• exact methods
• inequalities

Innovation

• extremal graph theory
• exact geometric approach
• convex hull of graph invariants
• interactive web tool
• non-isomorphic graph database