The identification number bound for connected identifiable graphs

Let $G$ be a connected identifiable graph of order $n\ge 2$ and maximum degree $\Delta\ge 2$. Write $\gamma^{{\rm ID}}(G)$ for the minimum size of an identifying code of $G$. Identification number bound. There exists a constant $c$ such that

$$\gamma^{{\rm ID}}(G) \le \left( \frac{\Delta - 1}{\Delta} \right) n + c.$$

This conjecture asks for the largest possible identification number of a connected identifiable graph in terms of its order and maximum degree. Earlier bounds showed that the identification number can be improved below $n$ by an amount depending on the maximum degree; the conjecture proposes the asymptotically sharp linear bound with an additive constant.