**Conjecture**A triangle-free graph with maximum degree has chromatic number at most .

This conjecture is a special case of Reed's , , and conjecture, which posits that for any graph, , where , , and are the clique number, maximum degree, and chromatic number of the graph respectively. Reed's conjecture is very easy to prove for complements of triangle-free graphs, but the triangle-free case seems challenging and interesting in its own right.

This conjecture is very much true for large values of ; Johansson proved that triangle-free graphs have chromatic number at most . Surprisingly, the question appears to be open for every value of greater than four, up until Johansson's result implies the conjecture.

Kostochka previously proved that the chromatic number of a triangle-free graph is at most , and he proved that for every there is a for which a graph of girth has chromatic number at most . Specifically, he showed that is sufficient. In [K] he posed the general problem: "To find the best upper estimate for the chromatic number of the graph in terms of the maximal degree and density or girth."

The conjecture is implied by Brooks' Theorem for . The three smallest open values of offer natural entry points to this problem. The easiest seems to be:

**Problem**Does there exist a -chromatic triangle-free graph of maximum degree 6?

Perhaps looking at graphs of girth at least five would also be a good starting point.

## Bibliography

[K] Kostochka, A. V., Degree, girth and chromatic number. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, pp. 679--696, Colloq. Math. Soc. János Bolyai, 18, North-Holland, Amsterdam-New York, 1978.

*[R] Reed, B.A., , and , J. Graph Theory 27 (1998) 177-212.

* indicates original appearance(s) of problem.