A distinguishing-index bound for graphs of minimum degree at least three

Let $G$ be a connected graph, and let $\Delta(G)$ and $\delta(G)$ denote its maximum and minimum degrees. A graph is $\delta$-minimally if it has the minimality property intended by the source. Its distinguishing index $D'(G)$ is the least number of labels in an edge labeling preserved only by the identity automorphism.

Proposed distinguishing-index conjecture. (i) If $G$ is a $\delta$-minimally graph with $\delta(G)\geq 3$ and $G$ is neither a complete bipartite graph nor a $\delta$-regular graph, then

$$D'(G)\leq \left\lceil \sqrt[\delta(G)]{\Delta(G)}\right\rceil.$$

(ii) If $G$ is connected and $\delta(G)\geq 3$, then

$$D'(G)\leq 1+\left\lceil \sqrt[\delta(G)]{\Delta(G)}\right\rceil.$$

The paper presents this as a conjecture because attempts to prove it had failed. The first part uses the source's term “$\delta$-minimally graph,” whose precise definition is not included in the supplied context.