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SUPER TOTAL LOCAL ANTIMAGIC COLORING OF GRAPHS

Combinatorics · math.CO · posed by Ravindra Pawar, Tarkeshwar Singh · open

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Statement

Every graph without isolated vertices admits super edge total local antimagic labeling.

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  • SUPER TOTAL LOCAL ANTIMAGIC COLORING 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.

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    NEW

    Problem: Let G=(V,E)G=(V,E) be a finite simple undirected graph, with n=∣V∣n=|V|, m=∣E∣m=|E|. A total labeling

    f:V∪E→{1,…,n+m}f:V\cup E\to \{1,\dots,n+m\}

    is a super edge total local antimagic labeling if

    f(E)={1,…,m},f(E)=\{1,\dots,m\},

    and for every edge uv∈Euv\in E, the vertex weights

    wf(v)=∑x∈N(v)f(x)+∑e∋vf(e)w_f(v)=\sum_{x\in N(v)} f(x)+\sum_{e\ni v}f(e)

    satisfy

    wf(u)≠wf(v).w_f(u)\ne w_f(v).

    The conjecture states: every finite simple graph without isolated vertices admits such a labeling.

    Result: The conjecture is true. In fact, the “without isolated vertices” hypothesis is not needed for the labeling-existence part.

    Proof. Choose any bijection

    g:E→{1,…,m}.g:E\to \{1,\dots,m\}.

    For each vertex vv, set

    s(v)=∑e∋vg(e),d(v)=∣N(v)∣.s(v)=\sum_{e\ni v} g(e),\qquad d(v)=|N(v)|.

    We shall choose a bijection p:V→{1,…,n}p:V\to\{1,\dots,n\}, then define

    f(e)=g(e)(e∈E),f(v)=m+p(v)(v∈V).f(e)=g(e)\quad(e\in E),\qquad f(v)=m+p(v)\quad(v\in V).

    Then

    wf(v)=∑u∈N(v)(m+p(u))+s(v)=md(v)+∑u∈N(v)p(u)+s(v).w_f(v)=\sum_{u\in N(v)}(m+p(u))+s(v) =md(v)+\sum_{u\in N(v)}p(u)+s(v).

    So it is enough to prove the following lemma.

    Lemma. Let C:V→RC:V\to\mathbb R be arbitrary. There exists a bijection p:V→{1,…,n}p:V\to\{1,\dots,n\} such that for every edge uvuv,

    ∑x∈N(u)p(x)+C(u)≠∑x∈N(v)p(x)+C(v).\sum_{x\in N(u)}p(x)+C(u)\ne \sum_{x\in N(v)}p(x)+C(v).

    Proof of lemma. Among all bijections p:V→{1,…,n}p:V\to\{1,\dots,n\}, choose one minimizing

    Φ(p)=2∑xy∈Ep(x)p(y)+2∑v∈VC(v)p(v).\Phi(p)=2\sum_{xy\in E}p(x)p(y)+2\sum_{v\in V}C(v)p(v).

    Suppose, for contradiction, that some edge uvuv has equal weights

    W(u)=W(v),W(u)=W(v),

    where

    W(z)=∑x∈N(z)p(x)+C(z).W(z)=\sum_{x\in N(z)}p(x)+C(z).

    Let p′p' be obtained from pp by swapping p(u)p(u) and p(v)p(v). Put

    δ=p(v)−p(u)≠0.\delta=p(v)-p(u)\ne 0.

    Only terms involving uu or vv change, and the edge term p(u)p(v)p(u)p(v) is unchanged. A direct computation gives

    Φ(p′)−Φ(p)=2δ(W(u)−W(v)−δ).\Phi(p')-\Phi(p) = 2\delta\bigl(W(u)-W(v)-\delta\bigr).

    Since W(u)=W(v)W(u)=W(v), this becomes

    Φ(p′)−Φ(p)=−2δ2<0,\Phi(p')-\Phi(p)=-2\delta^2<0,

    contradicting the minimality of pp. Hence no adjacent vertices have equal WW-values. The lemma follows.

    Now apply the lemma with

    C(v)=md(v)+s(v).C(v)=md(v)+s(v).

    Then for every edge uvuv,

    wf(u)=∑x∈N(u)p(x)+C(u)≠∑x∈N(v)p(x)+C(v)=wf(v).w_f(u)=\sum_{x\in N(u)}p(x)+C(u) \ne \sum_{x\in N(v)}p(x)+C(v)=w_f(v).

    Also f(E)={1,…,m}f(E)=\{1,\dots,m\}, so ff is a super edge total local antimagic labeling.

    Therefore every finite simple graph, in particular every graph without isolated vertices, admits a super edge total local antimagic labeling.

    Citation: No prior resolution is used here. The conjecture and definitions are from Pawar–Singh, “Super Total Local Antimagic Vertex Coloring of Graphs,” arXiv:2303.14019.

  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 proof is rigorous and addresses the correct conjecture. The minimization/swap lemma is valid: if an edge uvuv has equal induced weights, swapping p(u)p(u) and p(v)p(v) lowers Φ\Phi by −2(p(v)−p(u))2<0-2(p(v)-p(u))^2<0, contradicting minimality. Applying the lemma with C(v)=md(v)+∑e∋vg(e)C(v)=m d(v)+\sum_{e\ni v}g(e) gives a total bijection with f(E)={1,…,m}f(E)=\{1,\dots,m\} and distinct adjacent vertex weights. Searches found no prior exact resolution beyond the original arXiv paper.

    Novelty assessment

    TYPE1

    Classification rationale: The result appears genuinely new as stated, but it is a very short weighted/permutation variant of standard “lucky/additive” or local distance antimagic labeling ideas. The swap-minimization lemma is elegant but routine, and the conjecture is recent and niche. It would be best viewed as a short note/correction-style observation, not a substantial standalone combinatorics paper.

    Literature check: I found no prior paper proving Pawar–Singh’s Conjecture 2 (“every graph without isolated vertices admits a super edge total local antimagic labeling”) or the stronger arbitrary-offset lemma used here. Exact searches for “super edge total local antimagic,” “super total local antimagic,” and the conjecture wording led back to Pawar–Singh’s arXiv paper.

    Related literature is close but not the same: lucky/additive colorings consider neighbor-sum distinguishing vertex labels; local distance antimagic labeling uses a bijection V→{1,…,n}V\to\{1,\dots,n\} and neighbor sums; and dd-lucky labelings add the degree term. These do not seem to include arbitrary vertex offsets coming from fixed edge labels in the super edge total setting.

    Citation: Original conjecture: Ravindra Pawar and Tarkeshwar Singh, “Super Total Local Antimagic Vertex Coloring of Graphs,” arXiv:2303.14019.

    Related background: S. Czerwiński, J. Grytczuk, W. Żelazny, “Lucky labelings of graphs,” Information Processing Letters 109 (2009), 1078–1081; A.K. Handa, A. Godinho, T. Singh, “On local distance antimagic labeling of graphs,” AKCE Int. J. Graphs Comb. 21(1) (2023), 91–96.

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