For any fixed graph G, the subgraph isomorphism problem asks whether an n-vertex input graph has a subgraph isomorphic to G. A well-known algorithm of Alon et al. (J ACM 42(4):844–856, 1995. https://doi.org/10.1145/210332.210337) efficiently reduces this to the “colored” version of the problem, denoted \(G{\text{-}}{\mathsf {SUB}}\), and then solves \(G{\text{-}}{\mathsf {SUB}}\) in time \(O(n^{{\textit{tw}}(G)+1})\) where \({\textit{tw}}(G)\) is the treewidth of G. Marx (Theory Comput 6(1):85–112, 2010. https://doi.org/10.4086/toc.2010.v006a005) conjectured that \(G{\text{-}}{\mathsf {SUB}}\) requires time \(\varOmega (n^{{{\mathrm {const}}}\cdot {\textit{tw}}(G)})\) and, assuming the Exponential Time Hypothesis, proved a lower bound of \(\varOmega (n^{{{\mathrm {const}}}\cdot {\textit{emb}}(G)})\) for a certain graph parameter \({\textit{emb}}(G) \ge \varOmega ({\textit{tw}}(G)/\log {\textit{tw}}(G))\). With respect to the size of \({{\mathrm {AC}}}^0\) circuits solving \(G{\text{-}}{\mathsf {SUB}}\) in the average case, Li et al. (SIAM J Comput 46(3):936–971, 2017. https://doi.org/10.1137/14099721X) proved (unconditional) upper and lower bounds of \(O(n^{2\kappa (G)+{{\mathrm {const}}}})\) and \(\varOmega (n^{\kappa (G)})\) for a different graph parameter \(\kappa (G) \ge \varOmega ({\textit{tw}}(G)/\log {\textit{tw}}(G))\). Our contributions are as follows. First, we prove that \({\textit{emb}}(G)\) is \(O(\kappa (G))\) for all graphs G. Next, we show that \(\kappa (G)\) can be asymptotically less than \({\textit{tw}}(G)\); for example, if G is a hypercube then \(\kappa (G)\) is \(\varTheta \left( {\textit{tw}}(G)\big /\sqrt{\log {\textit{tw}}(G)}\right) \). This implies that the average-case complexity of \(G{\text{-}}{\mathsf {SUB}}\) is \(n^{o({\textit{tw}}(G))}\) when G is a hypercube. Finally, we construct \({{\mathrm {AC}}}^0\) circuits of size \(O(n^{\kappa (G)+{{\mathrm {const}}}})\) that solve \(G{\text{-}}{\mathsf {SUB}}\) in the average case, closing the gap between the upper and lower bounds of Li et al.