Dirac minimum-degree conjecture for rainbow connection

Determine whether every graph G with minimum degree at least half its order satisfies rc(G)≤2, equivalently whether every such graph admits a red-blue edge-coloring in which every non-adjacent vertex pair is joined by a two-edge path whose edges have different colors.

Background

The paper studies two-color rainbow connectivity under Dirac-type minimum-degree conditions. A graph has rainbow connection number at most two precisely when its edges can be colored red and blue so that every non-adjacent pair has a two-edge path using both colors.

The authors prove the assertion for several classes of Dirac and super-Dirac graphs, and asymptotically for almost all vertex pairs, but leave the general Dirac case unresolved.

References

Does every graph $G$ with $\delta(G)\geq v(G)/2$ satisfy $\mathrm{rc}(G)\le2$?

— Rainbow connecting $2$-colorings of super-Dirac graphs  (2609.11437 - Barát et al., 10 Sep 2026) in Question 1, Section 1 (Introduction)

It is not known whether there exists a constant $c>0$ such that $\delta(G)\ge v(G)/2+c$ implies $\mathrm{rc}(G)\le2$, and this is a compelling problem in itself.

— Rainbow connecting $2$-colorings of super-Dirac graphs  (2609.11437 - Barát et al., 10 Sep 2026) in Paragraph following Theorems 1.6 and 1.7, Section 1.1 (Results)

If $G$ is a $2n$-vertex Dirac graph that is either $\varepsilon$-close to $K_{n,n}$ or to $2K_n$, does $\mathrm{rc}(G)=2$ hold?

— Rainbow connecting $2$-colorings of super-Dirac graphs  (2609.11437 - Barát et al., 10 Sep 2026) in Question 4.1, Section 4 (Concluding remarks)

If $G$ is a Dirac graph that is a $(\nu,\tau)$-robust expander, for some suitable~$\nu$ and~$\tau$, does $\mathrm{rc}(G)=2$ hold?

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

Does every Dirac circulant (or Cayley) graph have an rc2-coloring?

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

Does every circulant (or Cayley) graph $G$ of minimum degree $v(G)/2+C$ for some constant $C$ have a generator-based rc2-coloring?

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

Let $G$ be a graph with $\delta(G)\ge (v(G)-1)/2$. Is it possible to 3-color the edges of $G$ so that every non-adjacent pair of vertices is connected by a non-monochromatic path of length two?

— Rainbow connecting $2$-colorings of super-Dirac graphs  (2609.11437 - Barát et al., 10 Sep 2026) in Question 4.2, Section 4 (Concluding remarks)