, By applying the transfer theorem (Theorem 2.6 in Section 2.3.2), we conclude that rt 3n spM g ptq´Sptq´ptq´S g ptqq{rt 3n sM g ptq

, According to our strategy, which is illustrated in Figure 6.8, the next step, which is represented by the only arrow between the two sides, will be using the unique embedding theorem of Robertson and Vitray (Theorem 6.2) to transfer the asymptotic enumeration of maps to graphs. However, Theorem 6.2 only applies to cubic graphs embeddable on S g that have facewidth at least 2g`32g`2g`3. Therefore, we need to control the facewidth of the embeddings of these cubic graphs

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