Determine the exact exponential growth constants for higher-treewidth graphs

Determine the exact values of the constants \(c(t)\), defined by the exponential growth rates of connected labeled graphs of treewidth at most \(t\), for integers \(t\ge 3\), in order to sharpen the corresponding planted-clique detection and recovery thresholds.

Background

The constants $c(t)$ govern the signal threshold $\lambda>1/\sqrt{c(t)}$ achieved by the paper’s bounded-treewidth signed-subgraph algorithms. The value $c(1)=e$ is known exactly, and $c(2)$ has a numerically characterized value, but the exact constants for larger treewidth are not available.

Consequently, the paper can derive only lower-bound-based algorithmic thresholds for $t\ge3$. Improved enumeration results, particularly an exact determination of $c(t)$, would directly yield sharper polynomial-time algorithms for detecting and recovering smaller planted cliques.