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Let $n$ be the minimum possible size of a 4-regular graph with girth $g$. Consider a graph with a minimum degree of at least four with girth $g$, can I say that the size of such graph must be at least $n$ (as each vertex has at least four neighbours, making the graph more denser than a 4-regular graph; hence, in order to obtain the same girth it requires more vertices) or there might be such a graph with size less than $n$?
Girth is the length of the shortest cycle in the graph.

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No, you can't say that, unless you prove it to be true. The way to know what you can say is to try to prove it; if you can prove it, you can say it, otherwise you can't.

It is possible that offering more freedom in the permissible degrees for each vertex allows constructing a smaller graph, so you can't trivially infer the conclusion you wish for.

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