In other words if a graph is 3-regular does it need to have 4 vertices? I ask because I have been asked to prove that if $n$ is an odd number and $G$ is an $n$-regular graph that $G$ must have an even number of vertices.
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It seems from your last sentence that you're asking whether an $n$-regular graph must have exactly $n+1$ vertices (rather than at least $n+1$ vertices). If so, as Gregor commented, the answer is no.
For the proof you're trying to find, try counting the number of incidences in two different ways.
I'm not sure how much detailed answer you want. So this is a hint, and the proof itself is hidden: Consider simple graph (no parallel edges, no loops on a vector) on $n$ vertices and think how many edges from a vertex can exist. As well, what if $n=0$?