Chromatic number, girth and maximal degree
It is proved that for every k≥4 there is a Δ(k) such that for every g there is a graph G with maximal degree at most Δ(k), chromatic number at least k and girth at least g. In fact, for a fixed k, the restriction of the maximal degree to Δ(k) does not seem to slow down the growth of the maximal girth of a k-chromatic graph of order n as n → ∞. © 1978.
Bollobás, B. (1978). Chromatic number, girth and maximal degree. Discrete Mathematics, 24 (3), 311-314. https://doi.org/10.1016/0012-365X(78)90102-4