Linear Time Subgraph Counting, Graph Degeneracy, and the Chasm at Size Six

We consider the problem of counting all $k$-vertex subgraphs in an input graph, for any constant $k$. This problem (denoted sub-cnt$_k$) has been studied extensively in both theory and practice. In a classic result, Chiba and Nishizeki (SICOMP 85) gave linear time algorithms for clique and 4-cycle counting for bounded degeneracy graphs. This is a rich class of sparse graphs that contains, for example, all minor-free families and preferential attachment graphs. The techniques from this result have inspired a number of recent practical algorithms for sub-cnt$_k$. Towards a better understanding of the limits of these techniques, we ask: for what values of $k$ can sub-cnt$_k$ be solved in linear time? We discover a chasm at $k=6$. Specifically, we prove that for $k < 6$, sub-cnt$_k$ can be solved in linear time. Assuming a standard conjecture in fine-grained complexity, we prove that for all $k \geq 6$, sub-cnt$_k$ cannot be solved even in near-linear time.