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\begin{document}
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\noindent Dear Editor,

\noindent   Thank you very much for your time on our manuscript, and we are grateful to the  referee 
for his or her  helpful comments as well.
According to the suggestions made by the reviewer,
we have revised the paper carefully.
Furthermoer, the current version is prepared in the E-JC style. 
 Please see our responses below to all the questions and comments.



\begin{enumerate}
\item  - p2, l9: The regular case was also proved in [2].

\textbf{Response:} The reference [2] has been added.

\item - p3, Lemma 2.1: This is an easy consequence of the K\H{o}nig-Hall theorem, I think it is slightly overweening to cite [4] as a source for the proof.

\textbf{Response:} We are not sure about  which theorem is the 
K\H{o}nig-Hall Theorem that the referee referrred to. But the proof of Lemma 2.1 is indeed an exercise applying Hall's Matching Theorem. Thus, we revise the paragraph prior to Lemma 2.1 as: We need the result below which guarantees a matching in a bipartite graph. It is an exercise to prove it by applying Hall's Matching Theorem.





\item - p5, l1: "...is odd and not equal to $1$ or $m+1$" -> Here $m+1$ is even (as $m$ is
odd), so the "$m+1$" part should be moved to the end of the next line.

\textbf{Response:} We rewrite "...is odd and not equal to $1$ or $m+1$; and $(b-i + 2,b -i + 1)$ if $i$ is even." as "... if $i$ is odd and not equal to $1$; and $(b-i + 2,b -i + 1)$ if $i$ is even and not equal to $m+1$.".




\item - p7, l2: trail -> trails.

\textbf{Response:} We have revised it.

\item - p12, l18: "Assume, w.l.o.g., that..." -> Why can we assume that? This is
indeed true, but a short explanation would be nice (the matching is fixed,
hence the order $x_1,...,x_m$ is also fixed, that's why it is not completely
trivial).



\textbf{Response:} We explained why we can assume that as follows in our paper:

 Under the above assignment
 of labels, we have that for any  $y_i, y_j\in \{y_1,y_2,\cdots, y_m\}$,  $\omega_H(y_i)=\omega_H(y_j)$.  Thus any assignment of
 distinct labels on the edges in $M$ results in
 a labeling of $G$ such that the sums of labels at vertices in $Y$ are all distinct.  Hence, we can choose an assignment of distinct labels on the edges in $M$ just based on the ordering of the values in $\{\omega_H(x_i)\,|\, 1\le i\le m\}$. 
 Therefore,  up to a reordering of edges in $M$, we may assume that $\omega_H(x_1)\le \omega_H(x_2)\le \cdots \le \omega_H(x_m)$.
 Now for each edge $x_iy_i\in M$, $1\le i\le m$,  assign the edge $x_iy_i$ with the label
 $i$.

\item - p13, l-10: maynot -> may not.

\textbf{Response:} We have revised it.


\item - p15, l12: Add a short explanation what "suppressing" means here.



\textbf{Response:} We added  an explanation. 


\item - References: Use the same style for all of the references (now, for example, the first one uses only initials of surnames).

\textbf{Response:} Revised correspondingly.


\end{enumerate}


\noindent Best regards!

\noindent Xiaowei Yu


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