Ore-type condition for two-color rainbow connection

Determine whether every n-vertex graph in which every pair of non-adjacent vertices u and v satisfies $\deg(u)+\deg(v)\ge n$ admits an rc2-coloring.

Background

The paper proves an rc2-coloring theorem under a stronger Ore-type condition, together with additional structural assumptions on common neighbors. It also recalls a bound of Dong and Li relating rainbow connection number to the minimum degree sum over non-adjacent pairs.

These results motivate the proposed strengthening of the Dirac question in which the degree-sum threshold replaces the minimum-degree condition.

References

Is it true that every $n$-vertex graph $G$ in which every pair $u,v$ of non-adjacent vertices satisfies $\deg(u)+\deg(v) \geq n$ has an rc2-coloring?

— Rainbow connecting $2$-colorings of super-Dirac graphs  (2609.11437 - Barát et al., 10 Sep 2026) in Question, Section 4 (Concluding remarks)