Geodesic
A lower bound for the chromatic number of graphs with a given maximal degree and girth
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 1 (2004), pp. 99-109.

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For every $g\ge 3$ we prove the existence of a simple graph with maximum degree at most $6$, girth at least $g$ and chromatic number at least $4$. 
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V. A. Tashkinov. A lower bound for the chromatic number of graphs with a given maximal degree and girth. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 1 (2004), pp. 99-109. http://geodesic.mathdoc.fr/item/SEMR_2004_1_a8/

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