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On the sum number and integral sum number of hypertrees and complete hypergraphs

Combinatorics · math.CO · posed by Martin Sonntag, Hanns-Martin Teichert · open

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Statement

Is it possible to prove an analogue of Theorem 3.1 for non-uniform hypertrees?

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  • On the sum number and integral sum number of hypertrees and complete hypergraphs
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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. exploration by a model · #1

    GPT-5.5 xhigh

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    Problem: Reconstructed conjecture: under the standard integral sum-hypergraph convention, every finite non-uniform hypertree TT has integral sum number ζ(T)=0\zeta(T)=0. That is, there should be an injective labeling ℓ:V(T)→Z\ell:V(T)\to\mathbb Z such that, for every subset X⊆V(T)X\subseteq V(T) with ∣X∣≥2|X|\ge2,

    X∈E(T)⟺∑x∈Xℓ(x)∈ℓ(V(T)).X\in E(T)\quad\Longleftrightarrow\quad \sum_{x\in X}\ell(x)\in \ell(V(T)).

    This is the direct non-uniform analogue of the cited uniform-hypertree theorem that uniform hypertrees are integral sum hypergraphs. The word “analogue” is ambiguous, but this is the natural zero-integral-sum-number interpretation.

    Result: The analogue is false.

    Let HH have vertices a,b,c,d,ea,b,c,d,e and edges

    E(H)={{a,b,c},{a,d},{d,e}}.E(H)=\{\{a,b,c\},\{a,d\},\{d,e\}\}.

    Its incidence graph is a tree, so HH is a hypertree. It is non-uniform since its edge sizes are 3,2,23,2,2.

    Assume, for contradiction, that HH has an integral sum labeling. Write

    A=ℓ(a), B=ℓ(b), C=ℓ(c), D=ℓ(d), E=ℓ(e).A=\ell(a),\ B=\ell(b),\ C=\ell(c),\ D=\ell(d),\ E=\ell(e).

    Because {a,d}\{a,d\}, {d,e}\{d,e\}, and {a,b,c}\{a,b,c\} are edges,

    P:=A+D,Q:=D+E,R:=A+B+CP:=A+D,\quad Q:=D+E,\quad R:=A+B+C

    must each be one of A,B,C,D,EA,B,C,D,E.

    Since {b,d}\{b,d\} and {a,b}\{a,b\} are nonedges, P≠A,DP\neq A,D; otherwise D=0D=0 or A=0A=0, creating an extra pair-edge. Hence

    P∈{B,C,E}.P\in\{B,C,E\}.

    Similarly Q≠D,EQ\neq D,E, so

    Q∈{A,B,C}.Q\in\{A,B,C\}.

    Also P≠QP\neq Q, since A+D=D+EA+D=D+E would force A=EA=E.

    By symmetry between bb and cc, the only cases are

    (P,Q)=(B,A),(B,C),(E,A),(E,B).(P,Q)=(B,A),(B,C),(E,A),(E,B).

    For each possible value of RR, the following table gives a forced extra edge; for example bc→dbc\to d means B+C=DB+C=D, so {b,c}\{b,c\} would be an edge.

    PQR=AR=BR=CR=DR=EBAbcd→dbc→dabd→dabce→aabcd→aBCbcd→dbc→dabd→dabce→cab→aEAbc→dac→dab→dbd→bbc→dEBbcd→dce→dab→eabce→bbc→d\begin{array}{c c|ccccc} P&Q& R=A& R=B& R=C& R=D& R=E\\ \hline B&A& bcd\to d& bc\to d& abd\to d& abce\to a& abcd\to a\\ B&C& bcd\to d& bc\to d& abd\to d& abce\to c& ab\to a\\ E&A& bc\to d& ac\to d& ab\to d& bd\to b& bc\to d\\ E&B& bcd\to d& ce\to d& ab\to e& abce\to b& bc\to d \end{array}

    None of the displayed subsets is one of {a,b,c},{a,d},{d,e}\{a,b,c\},\{a,d\},\{d,e\}. Thus every case creates a nonedge whose label-sum is a vertex label, contradiction.

    Therefore HH is a non-uniform hypertree with ζ(H)>0\zeta(H)>0, so the direct analogue of Theorem 3.1 fails.

    Citation: Original problem source: M. Sonntag and H.-M. Teichert, “On the sum number and integral sum number of hypertrees and complete hypergraphs,” Discrete Mathematics 236 (2001), 339–349, DOI: 10.1016/S0012-365X(00)00452-0. Counterexample above is self-contained.

  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 proposed counterexample attacks the natural analogue of Theorem 3.1, namely that every non-uniform hypertree has ζ(H)=0\zeta(H)=0. The hypergraph is indeed a non-uniform hypertree, and the case analysis for a putative integral sum labeling is exhaustive: the forced edge sums P=A+DP=A+D, Q=D+EQ=D+E, R=A+B+CR=A+B+C are constrained correctly, and each remaining case yields an additional subset whose label-sum is a vertex label but which is not an edge of HH. Hence no integral sum labeling exists.

    I found no evidence in the cited survey/search context of a prior published non-uniform hypertree counterexample or stronger resolution, so this should be treated as a valid new disproof of that direct analogue.

    Novelty assessment

    TYPE1

    Classification rationale: The accepted result is a very small counterexample to the broad “all non-uniform hypertrees have ζ=0\zeta=0” analogue. Even if new, it is a short finite obstruction with a case check, and it does not determine integral sum numbers or characterize non-uniform hypertrees. It would be useful as a remark or as part of a larger paper, but is not a standalone standard-journal contribution.

    Literature check: I found no prior source giving this counterexample or an equivalent disproof. Gallian’s 2023 Dynamic Survey lists the hypergraph results of Sonntag–Teichert: sum number results for hypertrees, ζ=0\zeta=0 for dd-uniform hypertrees, and bounds/conjectures for complete uniform hypergraphs, but no non-uniform hypertree resolution. Web searches for exact and variant phrases such as “non-uniform hypertree(s) integral sum”, “Problem 3.2 non-uniform hypertrees”, and “analogue of Theorem 3.1 non-uniform hypertrees” returned no relevant results. Broader searches found only the original Sonntag–Teichert papers and later work on complete hypergraphs/hypercycles, not the non-uniform hypertree analogue.

    Citation: M. Sonntag and H.-M. Teichert, “On the sum number and integral sum number of hypertrees and complete hypergraphs,” Discrete Mathematics 236 (2001), 339–349.
    J. A. Gallian, “A Dynamic Survey of Graph Labeling,” Electron. J. Combin. DS6, 26th ed., 2023, §7.1.

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