For a graph , we let denote the crossing number of , and we let denote the size of the largest complete subgraph of .

**Question**Does every graph with and have a 5-coloring?

In a nifty paper [OZ], Oporowski and Zhao get a little more mileage out of the classic Kempe-chain proof of the 5-color theorem, proving the following result.

**Theorem (Oporowski and Zhao)**Every graph with and is 5-colorable.

In particular, this shows that every graph with crossing number at most 2 is 5-colorable. The graph can be drawn in the plane with three crossings, so it is clear that the assumption is necessary in the highlighted theorem. However, it is possible that the assumption might be relaxed. The most extreme example known is the graph obtained from by removing the edges of a 5-cycle. This graph has , and , showing that the bound in the question is best possible.

Erman, Havet, Lidicky and Pangrac [EHLP] further improve this to

**Theorem (Erman et al.)**Every graph with and is 5-colorable.

On the other hand, the graph with , and (disproving the original conjecture) can be constructed in the following way: take two copies of with non-edges and , respectively. Let and be two triangles in these copies, and identify the corresponding vertices in these triangles. Finally, add the edge .

## Bibliography

*[OZ] B. Oporowski and D. Zhao, Coloring graphs with crossings.

[EHLP] R. Erman, F. Havet, B. Lidicky and O. Pangrac, 5-colouring graphs with 4 crossings, in preparation.

* indicates original appearance(s) of problem.