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All graphs here are finite and simple.

A $d$-factor of a graph is a spanning regular subgraph of degree $d$.

Where can I find theorems of this nature, for constants $a,b,c\gt 0$: If $G$ is a graph with all degrees in $[an,bn]$, and $d$ is an even integer in $[0,cn]$, then $G$ has a $d$-factor?

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To give at least a possible answer, and for those interested in doing a comprehensive literature search for an actual answer, it seems that the somewhat related survey

http://web.mat.bham.ac.uk/D.Kuehn/bcc09dkdo3.pdf

might be a helpful starting point.

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