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A survey of face-antimagic evaluations of graphs

Number theory · math.CO · math.NT · posed by Martin Bača, Edy T. Baskoro, Ljiljana Brankovic, Stanislav Jendrol', Yuqing Lin, Oudone Phanalasy, Joe Ryan, Andrea Semaničová-Feňovčíková, Slamin, Kiki A. Sugeng · open

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Statement

Find other possible values of the parameter dd and the corresponding dd-antimagic labeling of type (1,1,1)(1, 1, 1) for the generalized Petersen graph P(n,2)P(n, 2).

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  • A survey of face-antimagic evaluations of graphs
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  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: Reconstructed statement: for the standard planar embedding of the generalized Petersen graph P(n,2)P(n,2), find values of dd for which there is a type (1,1,1)(1,1,1) face dd-antimagic labeling, i.e. a bijection

    λ:V(P(n,2))∪E(P(n,2))∪F(P(n,2))→{1,…,∣V∣+∣E∣+∣F∣}\lambda:V(P(n,2))\cup E(P(n,2))\cup F(P(n,2))\to \{1,\dots,|V|+|E|+|F|\}

    such that the face weights

    w(f)=λ(f)+∑v∈∂fλ(v)+∑e∈∂fλ(e)w(f)=\lambda(f)+\sum_{v\in \partial f}\lambda(v)+\sum_{e\in \partial f}\lambda(e)

    form an arithmetic progression with common difference dd.

    The problem is open-ended rather than a yes/no conjecture. I resolve the existential reading by giving an explicit additional value: d=10d=10 for P(6,2)P(6,2).

    Result: Let P(6,2)P(6,2) have vertices ui,viu_i,v_i, outer edges ai=uiui+1a_i=u_iu_{i+1}, spokes bi=uivib_i=u_iv_i, and inner edges ci=vivi+2c_i=v_iv_{i+2}, with indices mod 66. Its faces are

    qi=uiui+1ui+2vi+2vi(0≤i≤5),q_i=u_i u_{i+1}u_{i+2}v_{i+2}v_i\quad (0\le i\le 5),

    and the two triangular inner faces

    T0=v0v2v4,T1=v1v3v5.T_0=v_0v_2v_4,\qquad T_1=v_1v_3v_5.

    Define λ\lambda by

    i012345λ(ui)3157613λ(vi)283138242636λ(ai)48921622λ(bi)152314101119λ(ci)272932353433λ(qi)121825172120\begin{array}{c|rrrrrr} i&0&1&2&3&4&5\\ \hline \lambda(u_i)&3&1&5&7&6&13\\ \lambda(v_i)&28&31&38&24&26&36\\ \lambda(a_i)&4&8&9&2&16&22\\ \lambda(b_i)&15&23&14&10&11&19\\ \lambda(c_i)&27&29&32&35&34&33\\ \lambda(q_i)&12&18&25&17&21&20 \end{array}

    and

    λ(T0)=30,λ(T1)=37.\lambda(T_0)=30,\qquad \lambda(T_1)=37.

    These labels are exactly 1,…,381,\dots,38. Direct calculation gives

    w(qi)=155+10i(0≤i≤5),w(q_i)=155+10i\quad(0\le i\le 5),

    and

    w(T0)=215,w(T1)=225.w(T_0)=215,\qquad w(T_1)=225.

    Thus the face weights are

    155,165,175,185,195,205,215,225,155,165,175,185,195,205,215,225,

    an arithmetic progression with common difference 1010. Hence P(6,2)P(6,2) admits a type (1,1,1)(1,1,1) face 1010-antimagic labeling.

    Citation: No external citation used for the construction above.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The construction is valid for P(6,2)P(6,2): the labels are exactly 1,…,381,\dots,38, matching ∣V∣+∣E∣+∣F∣=12+18+8|V|+|E|+|F|=12+18+8. Recomputing the face weights gives the six pentagonal weights 155,165,…,205155,165,\dots,205 and the two triangular weights 215,225215,225, so each face-size class has common difference d=10d=10 as required. This gives a legitimate additional dd-antimagic labeling of type (1,1,1)(1,1,1) for a generalized Petersen graph P(n,2)P(n,2). I did not find an existing literature source giving this specific d=10d=10 construction for P(6,2)P(6,2).

    Novelty assessment

    TYPE1

    Classification rationale: The accepted result is only a single explicit labeling for the small graph P(6,2)P(6,2) with d=10d=10. Even if new, it gives no infinite family, classification, or method beyond a finite certificate. It minimally answers an open-ended “find other values” problem and would not support a standalone combinatorics paper.

    Literature check: I found no prior source giving this exact d=10d=10 type (1,1,1)(1,1,1) face-antimagic labeling for P(6,2)P(6,2), nor a stronger theorem implying it. The 2017 survey records known values from Bača–Jendrol’–Miller–Ryan for P(n,2)P(n,2), including d=1d=1, d∈{0,2,3}d\in\{0,2,3\} in the n≡2(mod4)n\equiv2\pmod4 case, and d∈{2,3,6,9}d\in\{2,3,6,9\} in the n≡0(mod4)n\equiv0\pmod4 case, then leaves other values as Open Problem 1. Searches of open-access metadata and web sources for “face-antimagic”, “d-antimagic”, “P(n,2)”, “P(6,2)”, and “generalized Petersen” found only unrelated labeling variants or the cited survey/background papers.

    Citation: M. Bača et al., “A survey of face-antimagic evaluations of graphs,” Australas. J. Combin. 69(3) (2017), 382–393. Background: M. Bača, S. Jendrol’, M. Miller, J. Ryan, “Antimagic labelings of generalized Petersen graphs that are plane,” Ars Combin. 73 (2004), 115–128.

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