Three-Colorability of Continuous Planar Graphs

Prove or refute the conjecture that every continuous planar graph is (1/2,3)-colourable, and construct a continuous planar graph requiring at least three colours if the lower bound is attainable.

Background

Although every loopless planar combinatorial graph is four-colourable, the paper observes that Γ(K₄) is one-half, two-colourable, suggesting that continuous planar graphs may require fewer colors. The authors conjecture universal three-colourability and note that a formal example requiring at least three colors remains to be found or proved.

References

We conjecture that every continuous planar graph is $(\frac{1}{2},3)$-colourable. To get better intuition for this question, it would be helpful to find a continuous planar graph requiring at least $3$ colours and to provide a formal proof of this lower bound of $3$.

— Open Problems in Continuous Graphs  (2501.14554 - Grigoriev et al., 24 Jan 2025) in Section 3, subsection “Chromatic number,” paragraph “$(\frac{1}{2},3)$-colorability of continuous planar graphs”