Characterize attainable pairs of rainbow connection numbers

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

Background

The paper proves complete realizability results for the pairs (src(G),srcell(G)) and (rcell(G),srcell(G)), but does not obtain an analogous result for (rc(G),rcell(G)). The inequalities imply that any attainable pair must satisfy a=b=1 or 2 <= a <= b. For every pair with 2 <= a < b, the authors have not constructed a connected graph realizing rc(G)=a and rcell(G)=b. The paper also investigates sparse diameter-two graphs as possible candidates for separating the two parameters, but proves equality for the considered extremal family.

References

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 (Rainbow connection numbers with prescribed values)

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 3.4, Section 3 (following the proofs of Theorems 3.1 and 3.2)