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i read a note talking about this fact, bipartite graphs are rare in regular graphs. but it do not state how rare it is? just curious about it. thank you very much.

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1 Answer 1

Take $2n$ vertices. Regular bipartite graphs have $n$ vertices of each colour, and most have only one colouring because they are connected (for degree $\ge 3$). So take the asymptotic number of regular bipartite graphs, multiply by $\binom {2n}2$ graphs for the choice of colour classes, and divide by the asymptotic number of regular graphs. You can find all the necessary counts in many places such as this paper of Wormald.

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