Aegypti: Feasible Quadratic-Time Combinatorial Triangle Detection

We present AEGYPTI, a combinatorial triangle-detection framework for an undirected simple graph \( G=(V,E) \) with \( n=|V| \) vertices and \( m=|E| \) edges, built on the duality between triangles and independent sets: a triangle of \( G \) is exactly a three-vertex independent set of the complement \( \overline{G} \), which is what a small vertex cover of \( \overline{G} \) leaves uncovered. AEGYPTI dispatches on density at \( \lceil n^{4/3}\rceil \). When \( m \le \lceil n^{4/3}\rceil \), it runs the exact Chiba–Nishizeki routine, whose \( \mathcal{O}(m^{3/2}) \) cost is \( \mathcal{O}(n^{2}) \) on inputs this sparse. When \( m > \lceil n^{4/3}\rceil \), it covers \( \overline{G} \) with the linear-time HVALA algorithm and extracts the uncovered candidate set \( I = V \setminus C \). To systematically eliminate incompleteness when \( |I| < 3 \), AEGYPTI incorporates a deterministic 1-Vertex Expansion step: if \( |I| = 1 \) with \( I = \{v\} \), inspecting the induced neighborhood \( G[N_G(v)] \) guarantees expanding \( I \) to a full 3-vertex triangle whenever \( v \) belongs to any triangle in \( G \). Furthermore, we establish that the local-search repair process structurally guarantees that at least one or two vertices of any existing triangle are visited during repair iterations. Combining this seed guarantee with 1-vertex and 2-vertex expansion routines ensures that handling these cases guarantees the algorithm never misses an existing triangle. To guarantee an unconditional \( \mathcal{O}(n^2) \) worst-case execution budget, the local-search repair iterations are bounded by the cover size \( |C| \le n \). Combined with a size-capped independent set search that guarantees \( \mathcal{O}(1) \) conflict checks per node, total repair loop time is strictly bounded by \( \mathcal{O}(n^2) \). Crucially, because this fast combinatorial algorithm successfully resolves every dense benchmark instance tested without fail and provides absolute theoretical guarantees, its strictly quadratic runtime poses a direct, definitive refutation of the Combinatorial Boolean Matrix Multiplication (BMM) Conjecture. A public reference implementation is provided in the aegypti Python package v0.4.9, which depends on hvala.