Triangle Detection in H-Free Graphs

We initiate the study of combinatorial algorithms for Triangle Detection in $H$-free graphs. The goal is to decide if a graph that forbids a fixed pattern $H$ as a subgraph contains a triangle, using only "combinatorial" methods that notably exclude fast matrix multiplication. Our work aims to classify which patterns admit a subcubic speedup, working towards a dichotomy theorem. On the lower bound side, we show that if $H$ is not $3$-colorable or contains more than one triangle, the complexity of the problem remains unchanged, and no combinatorial speedup is likely possible. On the upper bound side, we develop an embedding approach that results in a strongly subcubic, combinatorial algorithm for a rich class of "embeddable" patterns. Specifically, for an embeddable pattern of size $k$, our algorithm runs in $\tilde O(n^{3-\frac{1}{2^{k-3}}})$ time, where $\tilde O(\cdot)$ hides poly-logarithmic factors. This algorithm also extends to listing all the triangles within the same time bound. We supplement this main result with two generalizations: 1) A generalization to patterns that are embeddable up to a single obstacle that arises from a triangle in the pattern. This completes our classification for small patterns, yielding a dichotomy theorem for all patterns of size up to eight. 2) An $H$-sensitive algorithm for embeddable patterns, which runs faster when the number of copies of $H$ is significantly smaller than the maximum possible $\Omega(n^k)$. Finally, we focus on the special case of odd cycles. We present specialized Triangle Detection algorithms that are very efficient: 1) A combinatorial algorithm for $C_{2k+1}$-free graphs that runs in $\tilde O(m+n^{1+2/k})$ time for every $k\geq2$, where $m$ is the number of edges in the graph. 2) A combinatorial $C_5$-sensitive algorithm that runs in $\tilde O(n^2+n^{4/3}t^{1/3})$ time, where $t$ is the number of $5$-cycles in the graph.