We revisit the well-studied problem of triangle count estimation in graph streams. Given a graph represented as a stream of $m$ edges, our aim is to compute a $(1\pm\varepsilon)$-approximation to the triangle count $T$, using a small space algorithm. For arbitrary order and a constant number of passes, the space complexity is known to be essentially ${\Theta(\min(m^{{3}/{2}}/T, m/\sqrt{T}))}$ (McGregor et al., PODS 2016, Bera et al., STACS 2017).
We give a (constant pass, arbitrary order) streaming algorithm that can circumvent this lower bound for low degeneracy graphs. The degeneracy, $\kappa$, is a nuanced measure of density, and the class of constant degeneracy graphs is immensely rich (containing planar graphs, minor-closed families, and preferential attachment graphs). We design a streaming algorithm with space complexity $\widetilde{O}(m\kappa/T)$. For constant degeneracy graphs, this bound is $\widetilde{O}(m/T)$, which is significantly smaller than both $m^{{3}/{2}}/T$ and $m/\sqrt{T}$. We complement our algorithmic result with a nearly matching lower bound of $\Omega(m\kappa/T)$