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Prime vertex labelings of several families of graphs

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prime-vertex-labelings-of-several-families-of-graphs-7Number Theorymath.COmath.NTposed by Nathan Diefenderfer, Dana C. Ernst, Michael G. Hastings, Levi N. Heath, Hannah Prawzinsky, Briahna Preston, Jeff Rushall, Emily White, Alyssa Whittemorerecorded: open · 1 machine check, unexamined

1 attempt · 1 machine check · no person has looked

Statement

Specifically, we conjecture that similar processes will work for cycle pendant stars up to stars of size 15.

Context

Candidate 7 of the open problems stated in "Prime vertex labelings of several families of graphs", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
    AI involvement
    ai discovered
    — the result was found by a model.
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    GPT-5.5 xhigh

    NEW

    Problem: Reconstructed statement: for the cycle pendant star

    Cn⋆P2⋆SmC_n\star P_2\star S_m

    with cycle vertices cic_i, spur vertices pip_i, and mm leaves adjacent to each pip_i, the intended conjecture is that the consecutive-block prime-labeling method used in the paper extends up to the Pillai barrier. Since each block has m+2m+2 vertices, Pillai’s 1717-consecutive-integers obstruction naturally gives the repaired precise form:

    ∀n≥3, ∀0≤m≤14,Cn⋆P2⋆Sm is prime.\forall n\ge 3,\ \forall 0\le m\le 14,\quad C_n\star P_2\star S_m \text{ is prime.}

    If “stars of size 15” is read as m=15m=15, the same block process is false: the aligned block

    {27830,27831,…,27846}\{27830,27831,\dots,27846\}

    has no element relatively prime to all other elements in the block, so no spur-center label can be chosen by that method.

    Result: Let L=m+2≤16L=m+2\le 16. Partition {1,…,Ln}\{1,\dots,Ln\} into consecutive blocks

    Bi={L(i−1)+1,…,Li}.B_i=\{L(i-1)+1,\dots,Li\}.

    Label

    f(ci)=L(i−1)+1.f(c_i)=L(i-1)+1.

    By Pillai’s theorem, every set of at most 1616 consecutive integers contains an element coprime to all the others. Moreover, in each BiB_i there is such an element not equal to its first element: if the first element is the only possible one, apply Pillai to the remaining L−1≤15L-1\le 15 consecutive integers; since the first element is coprime to all of them, the new element is also coprime to the whole block.

    Choose such an element of Bi∖{f(ci)}B_i\setminus\{f(c_i)\} for f(pi)f(p_i). Assign the remaining mm labels of BiB_i arbitrarily to the leaves adjacent to pip_i. Then pip_i is coprime to every adjacent label in its block.

    For cycle edges,

    gcd⁡(f(ci),f(ci+1))=gcd⁡(L(i−1)+1,Li+1)=1\gcd(f(c_i),f(c_{i+1}))=\gcd(L(i-1)+1,Li+1)=1

    because any common divisor divides LL and L(i−1)+1≡1(modL)L(i-1)+1\equiv 1\pmod L. Also gcd⁡(f(cn),f(c1))=1\gcd(f(c_n),f(c_1))=1. Thus all adjacent labels are relatively prime.

    Therefore Cn⋆P2⋆SmC_n\star P_2\star S_m is prime for every n≥3n\ge3 and 0≤m≤140\le m\le14.

    Audit: the proof uses exactly the cycle pendant star definition from the paper, labels all (m+2)n(m+2)n vertices bijectively with {1,…,(m+2)n}\{1,\dots,(m+2)n\}, checks all edge types, and introduces no extra graph hypotheses.

    Citation: S. S. Pillai, “On mm consecutive integers—III,” Proc. Indian Acad. Sci. 13 (1941), no. 6.
    N. Diefenderfer et al., “Prime Vertex Labelings of Several Families of Graphs,” Involve 9 (2016), 667–688.

    Reviews

    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE1

      PASS

      The proof correctly formalizes the paper’s “similar-process” range as Cn⋆P2⋆SmC_n\star P_2\star S_m with 0≤m≤140\le m\le 14, since each block has m+2≤16m+2\le16 vertices and the paper’s Pillai obstruction begins at m=15m=15. The labeling is bijective, checks all edge types, and Pillai’s theorem supplies the required spur label in each block. I found no later stronger/similar published resolution in the literature search.

      Novelty assessment

      TYPE1

      Classification rationale: The accepted result is, at best, a very short corollary of Pillai’s classical theorem on coprime representatives in blocks of at most 16 consecutive integers. It resolves only the repaired range m≤14m\le 14, not the ambiguous/literal “size 15” case, and adds essentially no new graph-labeling method beyond the block construction already suggested in the source paper. It is too small and routine for a standalone publication.

      Literature check: I checked the original arXiv/MSP/Involve record, OpenAlex metadata/citation data, and searched for the relevant phrases and notation: “cycle pendant star(s)”, “prime vertex labeling” + “cycle pendant”, Cn⋆P2⋆SmC_n\star P_2\star S_m, and “Pillai” + “prime vertex labeling”. I found no later paper or note explicitly proving this exact m≤14m\le 14 extension or a stronger result for these cycle pendant stars. The source paper remains the only located place stating the conjectural extension.

      Citation: Nathan Diefenderfer et al., “Prime vertex labelings of several families of graphs,” Involve 9 (2016), 667–688; arXiv:1503.08386.
      S. S. Pillai, “On mm consecutive integers—III,” Proc. Indian Acad. Sci. 13 (1941).

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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