The Local Antimagic Total Chromatic Number of Some Wheel-Related Graphs
Round 1
Reviewer 1 Report
The paper is clearly written and presents the new results which are justified in a convincing way. I only have some minor comments.
- "they [9] presented" or "they [11] determined" (page 2) - it is a strange localization of the references.
- In the formulation of the theorems the expressions "then we have", or just "we have", or "then" (without any "if") are unnecessary and, I suppose, have to be removed.
- Labelling of the vertices in the pictures is almost invisible (at least in the printout)
- What is "the sum of the weights of the vertex c" (proof of Theorem 2.5)? It seems that a vertex in a labelling has only one weight. Probably there should be "the sum of the weights of the vertex c and the edges incident with the vertex c".
- In Conclusion: "but instead of the local antimagic total chromatic coloring..." What is instead of it? This is not clear.
Author Response
Point 1: "they [9] presented" or "they [11] determined" (page 2) - it is a strange localization of the references.
Response 1: Change "they [9] presented" into "Putri et al. [9] presented"; Change "they [11] determined " into "Kurniawati et al. [11] determined ".
Point 2: In the formulation of the theorems the expressions "then we have", or just "we have", or "then" (without any "if") are unnecessary and, I suppose, have to be removed.
Response 2: Change “Then we have” into “Then” on page 13.
Point 3: Labelling of the vertices in the pictures is almost invisible (at least in the printout).
Response 3: Change the color of the vertices into white, and alter the size of the labellings of the vertices.
Point 4: What is "the sum of the weights of the vertex c" (proof of Theorem 2.5)? It seems that a vertex in a labelling has only one weight. Probably there should be "the sum of the weights of the vertex c and the edges incident with the vertex c".
Response 4: Change "the sum of the weights of the vertex c " into "the sum of the weights of the vertex c and the edges incident with the vertex c".
Point 5: In Conclusion: "but instead of the local antimagic total chromatic coloring..." What is instead of it? This is not clear.
Response 5: Change "but instead of the local antimagic total chromatic coloring of G" into "but may be the same in the local antimagic total chromatic coloring of G.”
Reviewer 2 Report
The authors study "The Local Antimagic Total Chromatic Number of Some Wheel-related Graphs". They bring a lot of new results. the paper is written very well. I did not find much mistakes of types. It is possible to read it easily and follow the text and proofs clearly way. The topic is still very alive between international community of graph theorists. Therefore I recommend the publication of the paper in the present form. I have jus one small recommendation: please include more new and recent results from the authors: Baca, Semancikova, Wang into Introduction part.
Author Response
Point 1: Please include more new and recent results from the authors: Baca, Semancikova, Wang into Introduction part.
Response 1: Add the following sentence after the second paragraph of the introduction part: “there are trees and graphs with vertices of even degrees and with chromatic index 3. From the results proved by Haslegrave [9], Baca et al. obtained that the local antimagic chromatic numbers of disjoint union of arbitrary graphs are finite if and only if none of these graphs contains an isolated edge as a subgraph.”
Author Response File: 
Author Response.docx
Reviewer 3 Report
In this paper, the authors investigate the local antimagic total chromatic number and precisely determine this property for specific families of graphs including the fan graph, the bowknot graph, the Dutch windmill graph, the analogous Dutch graph, and the flower graph. While most of the results appear to be new, the scope of the paper is very narrow and only includes computations for various families of graphs. These results could be the start of a paper, but I am unable to recommend the submitted manuscript for publication in its present form. I feel to be suitable for publication in Axioms, the results should provide results that are more general including new ideas and methods related to local antimagic total chromatic numbers. Here are some ideas: Looking at generalizations of the wheel graph and the Dutch windmill graph, what could be said about bi-regular graphs (graphs where the vertices have one of two different degrees)? For generalization of the analogous Dutch windmill graphs and the flower graphs, what could be said about tri-regular graphs (graphs where the vertices have one of three different degrees)? While a complete characterization of these properties would be ambitious, surely some progress could be made.
Major Comment:
According to http://matematika.fmipa.unand.ac.id/images/seminar-workshop/Pewarnaan%20Dan%20Pelabelan%20Graf/antimagiccoloring_unand.pdf, Theorem 2.3 was proved by Slamin, Dafik, Hasan (2018).
Minor Comments:
The vertex and edge labels on many of the graphs are very small. It would be helpful if they were larger.
Below is a small list of minor corrections. This list is almost surely partial and he paper should be carefully proofread to identify others.
Abstract: Third line from the end: "wheel-related graph" -> "wheel-related graphs"
Page 4: "The proof is completed." -> "The proof is complete." The same comment applies to the line above Figures 13 and 14 on page 11.
Page 6, sixth line from the end: "low bound" -> "lower bound"
Page 8, sentence below Figure 8: "by the table 1-5." -> "in Tables 1-5"
Page 9, last line: "Similarly, When", "Similarly, when"
Page 9, Example 3: "in Figure 9 and 10" -> "in Figures 9 and 10"
Page 9, fifth line from the end: "i∈ 1,2,⋯,n" -> "i∈ 1,2,…,n". Similar changes should be made in other parts of the paper.
Page 15: second line of the second paragraph: "more easy" -> "easier"
Author Response
Point 1: Abstract: Third line from the end: "wheel-related graph" -> "wheel-related graphs"
Response 1: Change "wheel-related graph" into "wheel-related graphs".
Point 2: Page 4: "The proof is completed." -> "The proof is complete." The same comment applies to the line above Figures 13 and 14 on page 11.
Response 2: Change "The proof is completed." into "The proof is complete." on page 4 and page 11.
Point 3: Page 6, sixth line from the end: "low bound" -> "lower bound".
Response 3: Change "low bound" into "lower bound" on page 6 and page 10.
Point 4: Page 8, sentence below Figure 8: "by the table 1-5." -> "in Tables 1-5"
Response 4: Change "by the table 1-5" into "in Tables 1-5" on page 8.
Point 5: Page 9, last line: "Similarly, When", "Similarly, when"
Response 5: Change "Similarly, When" into "Similarly, when" on page 9.
Point 6: Page 9, Example 3: "in Figure 9 and 10" -> "in Figures 9 and 10".
Response 6: Change "in Figure 9 and 10" into "in Figures 9 and 10" on page 9; and change "in Figure 2 and 3" into "in Figures 2 and 3" on page 4.
Point 7: Page 9, fifth line from the end: "i∈1,2,⋯,n" -> "i∈1,2,…,n". Similar changes should be made in other parts of the paper.
Response 7: Change "i∈1,2,⋯,n" into "i∈1,2,…,n" on page 9.
Point 8: Page 15: second line of the second paragraph: "more easy" -> "easier".
Response 8: Change "more easy" into "easier" on page 15.
Point 9: According to the PDF of Prof. Slamin, Theorem 2.3 was proved by Slamin, Dafik, Hasan (2018).
Response 9: Change “We obtain the exact value …… in the following theorem” into “ In fact, Slamin et al. [13] in 2018 and Amalia et al. [14] in 2021 had presented the exact value of the local antimagic total chromatic number of the fan graph. But the following local antimagic total labelling of the fan graph in Theorem 2.3 is different from that of these authors.” on Page 3.
Point 10: Here are some ideas: Looking at generalizations of the wheel graph and the Dutch windmill graph,what could be said about bi-reqular graphs (graphs where the vertices have one of two different degrees)? For generalization of the analogous Dutch windmill graphs and the flower graphs, what could be sald about tri-reqular graphs (graphs where the vertices have one of three different degrees)? While a complete characterization of these properties would be ambitious, surely some progress could be made.
Response 10: We are grateful to anonymous referees for good suggestions and references improving this article. Considering the local antimagic colorings of more general graphs, such as bi-reqular graphs and tri-reqular graphs are valuable suggestions, and giving the exact values of the local antimagic total chromatic numbers of these graphs would be ambitious in ten days, but we will put forward to it in the further research.
Point 11: The vertex and edge labels on many of the graphs are very small. It would be helpful if they were larger.
Response 11: The examples of graphs are small in order to illustrating the weights of the vertices and edges in fugures, and certainly larger numbers can be obtained from the formula. We will attempt to consider the larger graphs in furture works.
Author Response File: Author Response.docx
Round 2
Reviewer 3 Report
While the minor changes have been made there still is the major issue concerning the limited scope of the graphs in their paper. Ideas for possible generalizations were included in my last report. The authors should not feel restricted by the 10 days timeline. I would encourage them to take as much time as they need and then resubmit their article. The work the authors have done is commendable and with extensions of their results the paper will be better positioned for publication in Axioms.
This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.
