The well known result of Erdős, states that



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> Given integers $g > 2$ and $k > 1$ there exist a graph $G$ with $\chi(G) \geq k$ and girth at least $g.$
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What I am wondering is



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> When can we expect equality to hold? I.e for which parameters $(g,k)$ do we have graphs with girth $g$ and chromatic number $k$?
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Of course we cannot have $k=2$ and $g$ odd, but are there any other more interesting pairs $(g,k)$ for which there does not exist a graph of girth $g$ and chromatic number number $k$? In particular



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> Has anyone thought of searching/classifying these graphs in an analogous way as cages for girth and degree?
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After what Jacob said I propose the following



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> What is the smallest graph of girth $4$ and chromatic number $5.$
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Minimal graphs of prescribed girth and chromatic number
There are no more pairs (g,k). Indeed, one can start with a graph of large chromatic number and large girth. Deleting vertices one at a time, one gets a subgraph with chromatic number exactly k and girth at least g. As long as k>2, adding a disjoint cycle of length g keeps the chromatic number k and the girth will be g. For k=2 and g even, we can just take our graph to be a cycle of length g. 


