Finding Small Complete Subgraphs Efficiently
This paper addresses the efficient enumeration of small cliques—particularly triangles and $K_ell$ for $ell geq 3$—in sparse graphs. Methodologically, it introduces a minimalist triangle enumeration algorithm derived from first principles, achieving the $O(malpha)$ time bound, where $alpha$ denotes the graph’s arboricity; it also provides the first rigorous proof that the Chiba–Nishizeki algorithm attains a tight time lower bound for $K_ell$ enumeration, and establishes improved arboricity-sensitive counting and detection bounds for $ell geq 4$. Crucially, the approach abandons reliance on the Nash–Williams theorem, instead reconstructing a combinatorial framework grounded directly in arboricity properties. Theoretically, the results are optimal and asymptotically tight. Experimentally, the triangle enumerator significantly outperforms baselines, while for $K_ell$ with $ell geq 7$, it achieves consistent speedups on high-arboricity graphs.