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On the algorithmic complexity of adjacent vertex closed distinguishing colorings number of graphs

Combinatorics · math.CO · posed by Ali Dehghan, Mohsen Mollahajiaghaei · open

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Statement

For each positive integer tt, is there a bipartite graph GG such that V(G)=O(tc)V(G) = O(t^c) and dis[G]>tdis[G] > t, where cc is a constant number.

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  • On the algorithmic complexity of adjacent vertex closed distinguishing colorings number 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 every positive integer tt, there is a finite simple bipartite graph GtG_t with ∣V(Gt)∣=O(tc)|V(G_t)|=O(t^c), for an absolute constant cc, and dis⁡[Gt]>t\operatorname{dis}[G_t]>t.

    Here dis⁡[G]\operatorname{dis}[G] is the least kk such that some labeling f:V(G)→{1,…,k}f:V(G)\to\{1,\dots,k\} satisfies, for every edge uvuv with N[u]≠N[v]N[u]\ne N[v],

    ∑x∈N[u]f(x)≠∑x∈N[v]f(x).\sum_{x\in N[u]}f(x)\ne \sum_{x\in N[v]}f(x).

    The source’s V(G)=O(tc)V(G)=O(t^c) is naturally read as ∣V(G)∣=O(tc)|V(G)|=O(t^c).

    Result: Yes. One may take c=4c=4.

    Let N=2t+1N=2t+1. Let

    A1,…,ANA_1,\dots,A_N

    lie in one bipartition class and

    B1,…,BNB_1,\dots,B_N

    in the other. For every ordered pair (I,J)(I,J) of nonempty intervals of {1,…,N}\{1,\dots,N\}, add vertices xI,Jx_{I,J} in the AA-class and yI,Jy_{I,J} in the BB-class, and add edges

    xI,JyI,J,xI,JBi (i∈I),yI,JAj (j∈J).x_{I,J}y_{I,J},\qquad x_{I,J}B_i\ (i\in I),\qquad y_{I,J}A_j\ (j\in J).

    This graph is bipartite. Since there are N(N+1)/2N(N+1)/2 nonempty intervals,

    ∣V(Gt)∣=2N+2(N(N+1)2)2=O(t4).|V(G_t)|=2N+2\left(\frac{N(N+1)}2\right)^2=O(t^4).

    We prove no labeling f:V(Gt)→{1,…,t}f:V(G_t)\to\{1,\dots,t\} is closed distinguishing. Put

    ai=f(Ai),bi=f(Bi).a_i=f(A_i),\qquad b_i=f(B_i).

    Define prefix sums Ar=∑i=1raiA_r=\sum_{i=1}^r a_i and Bs=∑i=1sbiB_s=\sum_{i=1}^s b_i, with A0=B0=0A_0=B_0=0. The (N+1)2(N+1)^2 numbers Ar+BsA_r+B_s lie in the interval

    [0,AN+BN]⊆[0,2Nt],[0,A_N+B_N]\subseteq[0,2Nt],

    which contains only 2Nt+12Nt+1 integers. Since

    (N+1)2=(2t+2)2>2(2t+1)t+1=2Nt+1,(N+1)^2=(2t+2)^2>2(2t+1)t+1=2Nt+1,

    two distinct pairs (r,s)≠(r′,s′)(r,s)\ne(r',s') satisfy

    Ar+Bs=Ar′+Bs′.A_r+B_s=A_{r'}+B_{s'}.

    Strict monotonicity of the prefix sums gives r≠r′r\ne r' and s≠s′s\ne s'. Assume r>r′r>r'. Then necessarily s<s′s<s', and

    ∑i=r′+1rai=Ar−Ar′=Bs′−Bs=∑i=s+1s′bi.\sum_{i=r'+1}^{r} a_i=A_r-A_{r'}=B_{s'}-B_s=\sum_{i=s+1}^{s'} b_i.

    Thus there are nonempty intervals J,IJ,I such that

    ∑j∈Jf(Aj)=∑i∈If(Bi).\sum_{j\in J} f(A_j)=\sum_{i\in I} f(B_i).

    Now consider the edge xI,JyI,Jx_{I,J}y_{I,J}. Its closed-neighborhood sums are

    ∑z∈N[xI,J]f(z)=f(xI,J)+f(yI,J)+∑i∈If(Bi),\sum_{z\in N[x_{I,J}]} f(z) =f(x_{I,J})+f(y_{I,J})+\sum_{i\in I}f(B_i),

    and

    ∑z∈N[yI,J]f(z)=f(xI,J)+f(yI,J)+∑j∈Jf(Aj).\sum_{z\in N[y_{I,J}]} f(z) =f(x_{I,J})+f(y_{I,J})+\sum_{j\in J}f(A_j).

    These are equal. Also N[xI,J]≠N[yI,J]N[x_{I,J}]\ne N[y_{I,J}], since I,J≠∅I,J\ne\varnothing and one neighborhood contains BB-vertices while the other contains AA-vertices. Hence ff is not closed distinguishing.

    Therefore no labeling with labels in {1,…,t}\{1,\dots,t\} works, so

    dis⁡[Gt]>t.\operatorname{dis}[G_t]>t.

    Thus the conjectured polynomial-size bipartite examples exist, with ∣V(Gt)∣=O(t4)|V(G_t)|=O(t^4).

    Citation: Problem and definitions are from Dehghan and Molla Haji Aghaei, “On the algorithmic complexity of adjacent vertex closed distinguishing colorings number of graphs,” Discrete Applied Mathematics 218 (2017), 82–97. The construction 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 construction attacks the correct statement and is rigorous. For any labeling with colors in [t][t], the pigeonhole argument on the (N+1)2(N+1)^2 prefix-sum pairs forces equal sums over a nonempty interval of the AiA_i's and a nonempty interval of the BiB_i's. The corresponding gadget edge xI,JyI,Jx_{I,J}y_{I,J} then has equal closed-neighborhood sums while its endpoints have distinct closed neighborhoods, so no tt-color closed distinguishing labeling exists. The graph is bipartite and has O(t4)O(t^4) vertices.

    A literature check of the original paper/terminology and targeted searches did not reveal an existing polynomial-size bipartite construction solving this problem.

    Novelty assessment

    TYPE1

    Classification rationale: The result appears genuinely new for the stated bipartite adjacent closed-distinguishing problem, but it is a short elementary refinement of the exponential all-subsets construction in the source paper: replace all subsets by intervals and use a prefix-sum pigeonhole argument. It resolves a niche open problem, but likely only as a brief note/addendum or part of a collection of results, not a substantial standalone combinatorics paper.

    Literature check: The correct source is arXiv:1611.03181 / Discrete Applied Mathematics 218 (2017), not the arXiv id in input. That paper proves only an exponential-size bipartite construction, with ∣V∣=2t2+2(2t2−1)2|V|=2t^2+2(2^{t^2}-1)^2, and explicitly asks Problem 4 for polynomial size.

    I checked exact-title and terminology searches (“adjacent vertex closed distinguishing,” “closed distinguishing number,” “closed distinguishing labeling,” “dis[G] bipartite,” “V(G)=O(t^c),” related lucky/additive/sum-distinguishing terms), bibliographic mirrors, Scite/Semantic Scholar/OpenAlex citation lists, and forum-style searches. Later citations appear to be broad related work or different parameters. The closest related work is Axenovich–Caro–Yuster on sum-distinguishing sparse hypergraphs, which gives strong global closed-neighborhood distinguishing results for arbitrary/split graphs, but not the bipartite adjacent-vertex parameter in Problem 4.

    Citation: Ali Dehghan and Mohsen Mollahajiaghaei, “On the algorithmic complexity of adjacent vertex closed distinguishing colorings number of graphs,” Discrete Applied Mathematics 218 (2017), 82–97; arXiv:1611.03181. No prior polynomial-size bipartite resolution found.

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