The Steinburg conjecture, proposed by Richard Steinberg in 1976, is a problem in Graph Theory which states that:

“Every planar graph without 4-cycles and 5-cycles is 3-colourable.”

The property of a graph being 3-colourable means that it is possible to assign colours to its vertices so that no pair of adjacent vertices have the same colour, using only three different colours.

### Robert Steinburg

Robert Steinberg was born in 1922 and was a mathematician at UCLA. He made many important contributions to mathematics, including the Steinberg representation, the Lang-Steinberg theorem, the Steinberg group in algebraic K-theory and Steinberg’s formula in representation theory.

Notable Achievements:

- 1966: he was an invited speaker at the International Congress of Mathematics;
- 1985: won a Steele Prize, which is presented for distinguished research work and writing in mathematics;
- 1985: he was elected to the National Academy of Sciences;
- 1990: won the Jeffery-Williams Prize.

“I have had a good life.” – Robert Steinberg

### Counterexample

A paper, recently uploaded to the arXiv, by Vincent Cohn-Addad, Michael Hebdige, Daniel Kral, Zhentao Li and Esteban Salgado, shows the construction of a graph with no cycles of length 4 or 5 that isn’t 3 colourable. **This is a direct counterexample to the conjecture, hence proving it to be false.**

I had another post planned for today but I recently read this news and decided to make a post about it! Hope you enjoy. M x