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On the chromatic uniqueness of certain bipartite graphs

Combinatorics · math.CO · posed by Y.H. Peng · open

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Statement

Let GG and HH be two bipartite graphs. If G∼HG \sim H, prove (or disprove) that NK(m,n)(G)=NK(m,n)(H)N_{K(m,n)}(G)=N_{K(m,n)}(H) for all m,nm, n such that 2≤m≤n2 \le m \le n.

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  • On the chromatic uniqueness of certain bipartite 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 finite simple bipartite graphs G,HG,H, write G∼HG\sim H if they have the same chromatic polynomial. Let NK(m,n)(G)N_{K(m,n)}(G) be the number of subgraphs isomorphic to Km,nK_{m,n}. The claim asks whether

    G∼H  ⟹  NK(m,n)(G)=NK(m,n)(H)for all 2≤m≤n.G\sim H \implies N_{K(m,n)}(G)=N_{K(m,n)}(H) \quad\text{for all }2\le m\le n .

    Result: The statement is false.

    Let both graphs have bipartition

    X={x1,…,x5},Y={y1,…,y6}.X=\{x_1,\dots,x_5\},\qquad Y=\{y_1,\dots,y_6\}.

    Define GG by

    NG(y1)={x3,x4},NG(y2)={x1,x3,x4,x5},NG(y3)={x1,x3},NG(y4)={x1,x2,x3,x4},NG(y5)=NG(y6)={x1,x2,x4,x5}.\begin{aligned} N_G(y_1)&=\{x_3,x_4\},\\ N_G(y_2)&=\{x_1,x_3,x_4,x_5\},\\ N_G(y_3)&=\{x_1,x_3\},\\ N_G(y_4)&=\{x_1,x_2,x_3,x_4\},\\ N_G(y_5)&=N_G(y_6)=\{x_1,x_2,x_4,x_5\}. \end{aligned}

    Define HH by

    NH(y1)={x2,x3,x4,x5},NH(y2)={x2,x5},NH(y3)={x1,x3,x4},NH(y4)=X,NH(y5)={x1,x2,x3,x5},NH(y6)={x2,x4}.\begin{aligned} N_H(y_1)&=\{x_2,x_3,x_4,x_5\},\\ N_H(y_2)&=\{x_2,x_5\},\\ N_H(y_3)&=\{x_1,x_3,x_4\},\\ N_H(y_4)&=X,\\ N_H(y_5)&=\{x_1,x_2,x_3,x_5\},\\ N_H(y_6)&=\{x_2,x_4\}. \end{aligned}

    Both are finite simple bipartite graphs.

    For a bipartite graph with parts X,YX,Y, if Sj=N(yj)⊆XS_j=N(y_j)\subseteq X, then

    P(q)=∑π∈Π(X)(q)∣π∣∏j=16(q−ρπ(Sj)),P(q)=\sum_{\pi\in \Pi(X)} (q)_{|\pi|} \prod_{j=1}^{6}\bigl(q-\rho_\pi(S_j)\bigr),

    where ρπ(Sj)\rho_\pi(S_j) is the number of blocks of the partition π\pi meeting SjS_j. This counts colorings by first coloring XX.

    Applying this formula gives, for both GG and HH,

    PG(q)=PH(q)=q11−20q10+190q9−1115q8+4439q7−12443q6+24715q5−34125q4+31119q3−16778q2+4017q.\begin{aligned} P_G(q)=P_H(q) &=q^{11}-20q^{10}+190q^9-1115q^8+4439q^7\\ &\quad -12443q^6+24715q^5-34125q^4\\ &\quad +31119q^3-16778q^2+4017q. \end{aligned}

    Hence G∼HG\sim H.

    But their K3,3K_{3,3}-counts differ. A K3,3K_{3,3} corresponds to a triple in XX with at least three common neighbours in YY.

    In GG, the only such triples are

    {x1,x2,x4}↦{y4,y5,y6},{x1,x4,x5}↦{y2,y5,y6},\{x_1,x_2,x_4\}\mapsto \{y_4,y_5,y_6\}, \qquad \{x_1,x_4,x_5\}\mapsto \{y_2,y_5,y_6\},

    so NK(3,3)(G)=2N_{K(3,3)}(G)=2.

    In HH, the only such triple is

    {x2,x3,x5}↦{y1,y4,y5},\{x_2,x_3,x_5\}\mapsto \{y_1,y_4,y_5\},

    so NK(3,3)(H)=1N_{K(3,3)}(H)=1.

    Thus G∼HG\sim H, but

    NK(3,3)(G)≠NK(3,3)(H),N_{K(3,3)}(G)\ne N_{K(3,3)}(H),

    disproving the conjecture.

    Citation: No external citation; the counterexample is explicit 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 counterexample attacks the exact conjecture. The partition-coloring formula used for the chromatic polynomial is valid, and direct enumeration verifies that the two listed bipartite graphs have the same chromatic polynomial. The K3,3K_{3,3} counts also check out: GG has exactly two such subgraphs and HH exactly one, with no omitted triples contributing additional copies. Thus the conjecture is rigorously disproved. I found no prior matching result in the available literature search.

    Novelty assessment

    TYPE1

    Classification rationale: The accepted solution is a small explicit counterexample to Peng’s problem. It is useful as a negative answer, but the contribution is an ad hoc finite construction with direct verification and no broader theorem or new method. This is below the level of a standalone standard combinatorics paper.

    Literature check: I found the original problem in Peng’s paper and checked for the exact statement, the notation NK(m,n)N_{K(m,n)}, variants involving chromatically equivalent bipartite graphs and complete bipartite/biclique subgraph counts, and related chromatic-uniqueness literature on near-complete bipartite graphs. I found no published counterexample or stronger published resolution. Related works use such counts as invariants in restricted families, but do not settle the general question.

    Citation: Y.H. Peng, “On the chromatic uniqueness of certain bipartite graphs,” Discrete Mathematics 94 (1991), 129–140.

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