Aegypti: A Combinatorial Algorithm for Triangle Detection

We present AEGYPTI (v0.5.6), a combinatorial triangle-detection framework for an undirected simple graph \(G=(V,E)\) with \(n=|V|\) vertices and \(m=|E|\) edges. Execution is dynamically dispatched by density at the structural threshold \(\lceil n^{4/3}\rceil\). When \(m\le\lceil n^{4/3}\rceil\), an optimized Chiba–Nishizeki adjacency-intersection routine, sorted by non-decreasing degree, is invoked. For denser instances, the algorithm employs a randomized square-root partitioning strategy that isolates induced subgraphs and geometrically prunes intra-partition edges in \(\mathcal{O}(n^{2})\) time per iteration. Coupled with a linear-time Caro–Wei independent-set computation on the complement (and bipartite short-circuits), this strategy yields an expected runtime of \(\mathcal{O}(n^{2.5}\log n)\) for dense regimes. The randomness is drawn from a cryptographically secure source, guaranteeing a true Las Vegas algorithm whose expectation is taken solely over internal coin flips. We prove soundness, completeness, and the stated complexity bound in full detail. The resulting subcubic bound constitutes a randomized (Las Vegas) combinatorial counter-example to the generalized assumptions of the Combinatorial Boolean Matrix Multiplication (BMM) Conjecture. A public reference implementation is available in the aegypti Python package.