On the Chromatic Index of the Signed Generalized Petersen Graph GP(n,2)

Round 1

Reviewer 1 Report

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Author Response

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Reviewer 2 Report

The authors study the chromatic index of the signed generalized Petersen graph GP(n, 2).  Specifically, they argue the chromatic index is 3 for the cases where n = 3p (p odd) and n = 2p (p > 3).  I am not convinced by the proof of Theorem 1, as several cases rely on a claimed consequence of Lemma 15 that does not follow from its statement (see comments below).  In general, I believe significant improvements are also required to improve readability, particularly in Section 2, and verifying that all definitions and results introduced are necessary for this work.  I would reconsider after major revisions.  Detailed comments are provided below:

p1, l9: of signed graph -> of a signed graph

p1, l10:  satisfying that -> satisfying

p1, l11 (and throughout): Richard -> Behr

p1, l12 (and throughout): is consisted of -> consists of

p1, l18: with k < n/2, the generalized -> with k < n/2, then the generalized

p1, l20: v_{i + 2} -> v_{i + k}

p1, l21: Replace "We call the set... for l \in {1, 2, ..., n \gcd(n, k)}." with "The outer-cycle edges form the outer cycle, denoted c_0, and the inner-cycle edges form the inner cycles, denoted c_l, where l \in {1, 2, ..., gcd(n, k)} and |E(c_l)| = n \ gcd(n, k).  The spoke edges form a perfect matching, denoted M_s."

p2, l4: edge e, -> edge e;

p2, l12: every edges -> every edge

p2, l18: firstly considered -> first considered

p2, l18: Add a definition for signed graph (vertex) coloring

p2, l21 (and elsewhere): references to [9] appear to be superseded by references to [2] and do not appear to be necessary

p2, l24: A n-edge coloring -> An n-edge coloring

p2, l27: "The minimal number..." is not a complete sentence.

p2, l29: can receive -> receive

p2, l29 (and elsewhere): we call that -> we say that

p3, l16: different \pm a colors -> different \pm a colorings

p3, l20: did not affect -> does not affect

p3, l20: signed graph -> signed graphs

p3, l25: Two signed graph -> Two signed graphs

p3, l27: the matching of -> matchings of

p3, l28: matchings of generalized Petersen graph -> matchings of generalized Petersen graphs

p4, l1: in to two subsets -> into two subsets

p4, l1: M_1 and M_2 are introduced but do not appear to otherwise be used.

Figure 2.1: Structure C should not have a vertex in the middle of the inner cycle edge.

Figure 2.1(c) and (d) are never referred to.

Figures 2.1-2.4: edges in a perfect matching -> edges in the perfect matching

The paragraph after Proposition 6 should come before, as it introduces the notation used.

p4, l14: be as small as possible -> is minimized

p4, l15: then any of the following results holds -> then all of the following results hold

p5, l9: In describing the perfect matchings, I do not understand why A or C can be assumed to be consecutive.

Proposition 7 appears to be irrelevant; there does not appear to be anything "special" about the matchings indicated other than they are the ones used later, and other matchings are equally valid as described in the preceding paragraph.

Observations 8, 10, and 14 should be definitions.

The proofs of Propositions 9, 11, 12, and 13 do not add to my understanding.  I believe in each of these cases, the cycles could simply be presented; it should be easily verified that precisely the required edges are used.

Observation 10 should have subcases indented to improve readability.

If no spoke edge is negative, then each of |E_s^\sigma-(M_C_i^p)| and |E_s^\sigma-(GP_\sigma(n, 2))| have the same parity, so the statement following Lemma 15 cannot be concluded.

p11, l24: is on outer cycle -> is on the outer cycle

p12, l16 (and elsewhere): generalized Petersen graph -> the generalized Petersen graph

p12, l17 (and similarly elsewhere): have a outer cycle and a inner cycle -> has an outer cycle and an inner cycle

Claim 18, Case 2, Case 3.1, and Case 3.2 do not follow from Lemma 15 (as stated).

p14, l25 (and similarly elsewhere): E(T_{i-3}), E(T_{i}), and E(T_{i + 3}) should be simplified by writing E(T_l).

p14, l29 (and similarly elsewhere): T_1, T_2, and T_3 need replacing with their proper forms.

Ref [2]: Edge coloing -> Edge coloring

Ref [7]: P. Diestel -> R. Diestel

Author Response

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Reviewer 3 Report

● Page 1, Sect. 1: do you give the definition for a general k or for k=2?

● Page 2, line 1-2: can you uniform notations and write better? Also line 3.

● The English language must be revised in the whole paper. Also pay attention to the use and non-use of the article and to expressions not suitable for English.

● Sect. 1: try to give some more definitions, or more explanations, when quotes a result.

● The bibliography is seriously insufficient. Only two articles from the 1960s appear, two books, some papers submitted or published by some of the authors and little else. It is important to give the reader more information on current research and, if possible, to highlight links to your own. For example, from double generalized Petersen graphs to unit distance graphs, from enumeration of cycles to groups on GPG, ecc.: there are many related works. You can also refer to recent papers that connect labeled graphs with unimaginable numbers {doi: 10.24330/ieja.1058413} or with GPG themselves {doi: 10.1016/j.ipl.2013.03.017}, etc. Continue like this.

● Page 4, line 1: into.

● Pay attention to spaces, especially before and after the brackets, in the whole paper.

● Page 5: there are some unnecessary capitalizations.

● Page 7, line 4: paly.

● Page 9, line -5: "the case $i=1$". Check elsewhere.

● Page 13: "By Lemma 15", "Propisition 12".

● Page 15, line 16: a couple of round brackets is superfluous.

● For the next submission, I recommend to use the MDPI Latex template.

 

Author Response

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Round 2

Reviewer 2 Report

I thank the authors for their modifications to the paper, which has much improved.  Based on the additional details in Theorem 1, I believe the results to be correct.  I think some proofs in Section 2 could still be simplified (further details below), but will not insist on it and appreciate the addition of the explicit Hamilton cycles.  A few minor edits are also required.

With regards to point 12, referring to the results of Zhang et al. prior to the results of Behr, and adding the fact that the definitions were introduced independently addresses my concerns regarding this point.

With regards to point 24, reframing the matchings as Proposition 5 addresses my concerns.

With regards to point 31, my concerns would be addressed by the phrasing "in what follows, we will only consider the matchings in which the structures A are consecutive".

I reiterate my point 32, the only thing "special" about the matchings is that they are used to prove later results.  The word "special" here has no meaning and should not be used.  I suggest, "For proving later results, we will make use of the following matchings"

With regards to point 34, which now refers to Propositions 4-7, I still believe the proofs are unnecessarily detailed, and that most details could be omitted to be verified by the reader.  In particular, the division of edges into outer, inner, and spoke here seems to be largely irrelevant.  The key point is to demonstrate the Hamilton cycle by stating it explicitly, as has been added.

With regards to point 35, it is still difficult to distinguish at a glance when moving from one case to the next; either increase the indentation within each case or add line breaks between the cases.

Line 39: E(G) -> {+1, -1}. The pair -> E(G) -> {+1, -1}, the pair

Line 52: coloring of signed graph -> coloring of signed graphs

Line 52: was fist considered -> was first considered

Line 53: to color set -> to a color set

Line 53, 54: vertices jointed by -> vertices joined by

Line 56: chromatic number of signed graph -> chromatic number of a signed graph

Line 57: Brooks' theorem of signed graph -> Brooks' Theorem for signed graphs

Line 88: the signed graph -> if a signed graph

Lines 90, 91: 3-coloring -> Tait coloring

- Additionally, Tait coloring ought to be defined.

Lines 93, 98: Results should be specified rather than just saying "got some results".

Line 109: some notations -> some notation

Line 197: In Figure 2, We -> In Figure 2, we

Line 380: have an outer cycle -> has an outer cycle

Line 420: v_j v_{j + 2} -> As v_j v_{j + 2}

Line 549: tait colorings -> Tait colorings

Author Response

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Reviewer 3 Report

The manuscript has undergone a good process of improvement and revision. However, the bibliography remains very essential and the graphics should be improved before publication, perhaps with the help of the editorial office. The paper would therefore only need a limited revision.

Author Response

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