The Decomposability of Toroidal Graphs without Adjacent Triangles or Short Cycles

Round 1

Reviewer 1 Report

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Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

In the paper under review, the authors proved that every toroidal graph without adjacent triangles is (3, 1)-decomposable, and for {i, j} ⊆ {3, 4, 6}, every toroidal graph G without cycles of length i or j is (2, 1)-decomposable; see Theorems 2.1 and 3.1. In this reviewer's opinion, these results are interesting, new, and correct. I recommend the publication of this paper. Some minor comments are as follows that the authors may consider for revising their paper.

1. I suggest revising Line 37. One of the possibilities is "Is it possible to reduce the maximum degree of H in (*) to 5 or 4? This question remains open".

2. Revise "describe a discharging procedure" in Line 73; what do you mean by a discharging procedure?

3. I suggest inserting the text "Proof:" right before Line 72, removing the text "Lemma 2.1" and "Proof" in Lines 75 and 79, respectively, and revising the resulting proof of Theorem 2.1 accordingly. I suggest revising the proof of Theorem 3.1 in a similar way.

4. Revise "counterexample of 3.1" as "counterexample of Theorem 3.1"

 

  

 

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

The manuscript contains a valuable work about decomposability of toroidal graphs. The authors prove that prove that every toroidal graph without adjacent triangles is in a certain manner decomposable. Some additional coloring lemmas are also provided.

In my opinion the literature survey is a little abrupt since the journal is of general mathematics and the manuscript if of specialized field mathematics. Besides, the references list is extremely short. I would like to see smoothly  introduced in a paragraph or two general concepts on different types of graphs. Also I would like to see mentioned some important applications of graph theory in general as well as some of the applications of the selected topic.

In my opinion the publisher have a good collection of works related to the graph theory in general and to the colorability in special and I am sure that the authors can make a properly introduction based on those works for the people less familiarized with theoretical studies about the importance and the use of the graphs (for instance in molecular sciences) as well as a proper discussion about what can others use from the reported results to solve other challenges.

Also in my opinion some used terminology should be better introduced (how short is short? for instance).

Overall, the manuscript is valuable and deserves publication, but as I mentioned above, it needs a little more preparatory part as well as some discussions and even some conclusions.

 

Author Response

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Author Response File: Author Response.pdf