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I would like to thank the referee for their helpful comments and recommendations; I have indicated below how I have addressed them.

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\item \textit{The construction of the directed graphs in Section 2.3 has previously been studied. In arXiv:1311.2783v3, these graphs are called `plane alternating dimaps', and this paper describes the history of the topic. It seems these objects were first discovered by Tutte (The dissection of equilateral triangles into equilateral triangles, \textit{Proc. Cambridge Philos. Soc.} \textbf{44} (1948) 463--482). I recommend that this connection be included in Section 2.3.}

I have added discussion of this connection into both Section 2.2 and Section 2.3. See second last paragraph on Page 5; last paragraph on Page 6; third last paragraph on Page 7; and lines $-5$ and $-4$ on Page 15.

\item \textit{The proof of Proposition 3 seems to leave many details to the reader. I would recommend either adding a picture to illustrate the arguments or adding more details to the proof.}

I have added figures illustrating each of the three cases (Figures 4, 5 and 6 in the updated version).

\item \textit{In the proof of Proposition 4, a closed curve $L$ is defined, but is referred to as `this loop' in the following sentence (change to `this curve' or simply $L$).}

I have made the suggested change.

\item \textit{Typo in the proof of Proposition 4: `back face' should be `black face'.}

I have corrected the typo.

\item \textit{In the proof of Proposition 4, `the two faces containing $e_1$' can be ambiguous; I suggest explicitly stating that these faces are in $\mathcal{G}$ rather than in $D_K$, and specifying which one is black/white.}

I have made the suggested change and also included a figure illustrating this case (Figure 7 in the updated version).

\item \textit{Please check the grammar of the sentence following the proof of Proposition 4 (on page 9).}

I have corrected the grammar.

\item \textit{On page 11, the last sentence of the second paragraph (`Thus, although the beginning \dots') appears to be missing some commas and is difficult to read.}

I have reworded the sentence so that it now reads: ``The beginning of our argument follows that of [?] (in setting up the use of a refinement of Suen's Inequality); however, as we are interested in the growth rate as the number of arcs increases, the remainder necessarily follows a different approach.''


\item \textit{In the proof of Theorem 8 on page 12, the graph $\Gamma$ is defined on the vertex set $\mathcal{D}_I$, which has not yet been defined. I believe $\Gamma$ should have vertex set $\mathcal{D}_{I-i_0}$. Also, in the definition of the edges of $\Gamma$, I suggest stating explicitly that $\alpha$ and $\beta$ are directed cycles (two vertices sharing a vertex sounds a bit awkward).}

I corrected the vertex set to be ``$\mathcal{D}_{I-i_0}$''. I have also reworded the remainder of the sentence in the manner suggested. It now reads ``Define a graph $\Gamma$ on the vertex set $\mathcal{D}_{I-i_0}$, with an edge between the vertices in $\Gamma$ corresponding to the cycles $\alpha$ and $\beta$ of $D_I$ if and only if $\alpha$ and $\beta$ share a vertex in $D_I$.''.


\item \textit{In the proof of Theorem 8 on page 12, it is stated that $i_0$ can be chosen in such a way to simplify the inequality. I did not understand this step; please provide details of how $i_0$ is chosen.}

I have added more details showing this (Lemma 9 in the updated version). While doing so I noticed that I could strengthen the result to improve the factor of 1/2 to 1 in Equation (2). So I have also made this change. 

It does not impact on the result of Theorem 8 (Theorem 9 in the updated version) but it does substantially improve the result of Theorem 9 (Theorem 10 in the updated version). I have corrected the factors of 1/2 throughout the proofs of both these theorems.


\item \textit{In the proof of Theorem 8 on page 13, there are summations over $v\in V$, but $V$ has not been defined. Should this be $R\cup C\cup S$?
}

While addressing the previous comment I have added the definition of $V$.

\item \textit{In the proof of Theorem 9 on page 14, we are asked to consider a vertex $v\in N_4\cup N_6$. This vertex is not used anywhere else in the proof.}

I have removed this sentence.


\item \textit{In the proof of Theorem 9 on page 15, it is stated that the maximum value of $\min\{\dots,\dots\}$ occurs when $n_6/n=3/5$. I recommend also stating that this maximum value is $\frac{3}{5}(\ln 3 +\ln 2)\,n$.}

I have updated the sentence to read: ``$\ldots$ occurs when $n_6/n=3/5$; yielding $3\ln(\mathcal{T}(D_I))<\frac{3}{5}(\ln(3)+\ln(2))n=\frac{3}{5}\ln(6)n$.''.
\end{enumerate}

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