Characterize attainable pairs of rainbow connection and list rainbow connection numbers

Characterize all pairs of positive integers (a,b) for which there exists a connected graph G satisfying rc(G)=a and rc^ℓ(G)=b.

Background

The inequalities rc(G)≤rcℓ(G), together with the characterization of graphs having rc(G)=1, imply that any attainable pair must satisfy either a=b=1 or 2≤a≤b. The paper establishes analogous complete characterizations for pairs involving src(G),srcℓ(G) and rcℓ(G),srcℓ(G), but does not obtain such a result for rc(G),rcℓ(G). In particular, for every pair 2≤a<b, the authors report that they have not constructed a connected graph with rc(G)=a and rcℓ(G)=b.

References

For the remaining inequality $rc(G)\le rc\ell(G)$ of (\ref{ineqs}), we have not been able to prove a similar result. We propose the following problem. Characterise all pairs of positive integers $a$ and $b$ such that, there exists a connected graph $G$ with $rc(G) = a$ and $rc\ell(G) = b$. Is it true that $rc(G)=rc\ell(G)$ for all connected graphs $G$?

— List rainbow connection number of graphs  (2503.08964 - Tang et al., 12 Mar 2025) in Problem rcrclcompprob, Section 3, immediately following the proofs of Theorems srcsrclcompthm and rclsrclcompthm