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Some families of chromatically unique bipartite graphs

Combinatorics · math.CO · posed by Xiebin Chen · open

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Statement

For every r ≥0 and every h≥h0(r), where h0(r) depends on r, there is an integer p1(h,r) depending on h and r, such that for every p≥p1(h,r), each member of K^{-r}(p,p+h) is χ-unique.

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  • Some families of chromatically unique bipartite graphs
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  1. exploration by a model · #1

    GPT-5.5 xhigh

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    NEW

    Problem: Reconstructed statement: in finite simple graphs, K−r(p,q)K^{-r}(p,q) denotes the family of graphs obtained from Kp,qK_{p,q} by deleting exactly rr edges. A graph is χ\chi-unique if every graph with the same chromatic polynomial is isomorphic to it. The conjecture asserts

    ∀r≥0 ∃h0(r) ∀h≥h0(r) ∃p1(h,r) ∀p≥p1(h,r),\forall r\ge 0\ \exists h_0(r)\ \forall h\ge h_0(r)\ \exists p_1(h,r)\ \forall p\ge p_1(h,r),

    every member of K−r(p,p+h)K^{-r}(p,p+h) is χ\chi-unique. This is the standard meaning of the notation in this bipartite chromatic-uniqueness context.

    Result: The conjecture is false. A counterexample exists already for r=8r=8, for every h≥1h\ge1 and all sufficiently large pp.

    Let q=p+hq=p+h. In Kp,qK_{p,q}, delete edges according to the following 4×44\times4 00-11 matrix, with rows in the pp-part and columns in the qq-part:

    D=(1010101101011000).D= \begin{pmatrix} 1&0&1&0\\ 1&0&1&1\\ 0&1&0&1\\ 1&0&0&0 \end{pmatrix}.

    Let D⊤D^\top be its transpose. Embed both in the first four vertices of each part, and define

    G=Kp,q−D,G′=Kp,q−D⊤.G=K_{p,q}-D,\qquad G'=K_{p,q}-D^\top .

    Both lie in K−8(p,q)K^{-8}(p,q).

    For any deleted-edge graph D⊆Kp,qD\subseteq K_{p,q}, repeated deletion-contraction gives

    P(Kp,q−D,t)=∑S⊆E(D)P(Kp,q/S,t).P(K_{p,q}-D,t)=\sum_{S\subseteq E(D)} P(K_{p,q}/S,t).

    A subset SS contributes nonzero precisely when each nontrivial component of SS is a complete bipartite graph. If such an SS covers aa left vertices and bb right vertices and has cc nontrivial components, then

    P(Kp,q/S,t)=(t)c P(Kp−a,q−b,t−c).P(K_{p,q}/S,t)=(t)_c\,P(K_{p-a,q-b},t-c).

    Thus the chromatic polynomial depends only on the counts ND(a,b,c)N_D(a,b,c) of such admissible subsets.

    For the above DD, the nonzero counts are:

    (a,b,c)ND(a,b,c)(0,0,0)1(1,1,1)8(1,2,1),(2,1,1)5(1,3,1),(3,1,1)1(2,2,1)1(2,2,2)18(2,3,2),(3,2,2)15(2,4,2),(4,2,2)3(3,3,2)8(3,3,3)12(3,4,2),(4,3,2)1(3,4,3),(4,3,3)8(4,4,3)4(4,4,4)1.\begin{array}{c|c} (a,b,c)&N_D(a,b,c)\\ \hline (0,0,0)&1\\ (1,1,1)&8\\ (1,2,1),(2,1,1)&5\\ (1,3,1),(3,1,1)&1\\ (2,2,1)&1\\ (2,2,2)&18\\ (2,3,2),(3,2,2)&15\\ (2,4,2),(4,2,2)&3\\ (3,3,2)&8\\ (3,3,3)&12\\ (3,4,2),(4,3,2)&1\\ (3,4,3),(4,3,3)&8\\ (4,4,3)&4\\ (4,4,4)&1 . \end{array}

    The table is symmetric in a,ba,b. Since transposition sends ND(a,b,c)N_D(a,b,c) to ND⊤(b,a,c)N_{D^\top}(b,a,c), we get

    ND(a,b,c)=ND⊤(a,b,c)N_D(a,b,c)=N_{D^\top}(a,b,c)

    for all triples. Hence

    P(G,t)=P(G′,t).P(G,t)=P(G',t).

    But G≇G′G\not\cong G' when p≥7p\ge7 and q>pq>p. Indeed both graphs are connected bipartite, so an isomorphism must preserve the two part sizes. It would therefore induce a side-preserving isomorphism between DD and D⊤D^\top. No such isomorphism exists: in DD, the unique left vertex of deletion-degree 11 is adjacent to the unique right vertex of deletion-degree 33; in D⊤D^\top, the unique left vertex of deletion-degree 11 is adjacent to a right vertex of deletion-degree 22.

    Thus for every h≥1h\ge1 and every p≥7p\ge7, K−8(p,p+h)K^{-8}(p,p+h) contains a graph that is not χ\chi-unique. Therefore no h0(8)h_0(8) can satisfy the conjecture.

    Citation: No external resolution is used; the counterexample above is explicit.

  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 TYPE2

    PASS

    The disproof is mathematically sound. The deletion–contraction expansion is correctly specialized: only subsets whose components are complete bipartite contribute, and their contribution depends only on (a,b,c)(a,b,c). The stated finite count table for the chosen 8-edge deletion pattern is symmetric, so Kp,q−DK_{p,q}-D and Kp,q−D⊤K_{p,q}-D^\top have the same chromatic polynomial. For q>pq>p, connectedness forces any isomorphism to preserve bipartition sizes, and the deletion patterns are not side-preservingly isomorphic, so the graphs are nonisomorphic. Thus the conjecture is false for r=8r=8.

    A literature search found related small-rr uniqueness papers but no prior similar counterexample or stronger disproof.

    Novelty assessment

    TYPE2

    Classification rationale: The result appears genuinely new and gives an explicit counterexample to a published conjecture, already with r=8r=8. Although the proof is short and computational/finite in flavor, it resolves the conjecture negatively and would plausibly support a short standalone note in a standard graph theory/combinatorics journal. It is not TYPE3: the conjecture is fairly specialized and the method is not a broad major advance.

    Literature check: I found no prior occurrence of this counterexample or any stronger disproof. Searches of scholarly indexes and web results for Chen’s conjecture, K−r(p,q)K^{-r}(p,q), K−8K^{-8}, h0(r)h_0(r), p1(h,r)p_1(h,r), and “chromatically unique complete bipartite graphs with edges deleted” led only to positive/partial uniqueness papers. In particular, Dong–Koh–Teo–Little–Hendy and Roslan/Hasni–Peng prove uniqueness for various structured deletion graphs or large maximum-degree cases, but they do not cover this 88-edge, maximum deletion-degree 33 nonunique construction.

    Citation: No prior citation found for the counterexample. Relevant literature includes X. Chen, “Some families of chromatically unique bipartite graphs,” Discrete Math. 184 (1998), 245–252, doi:10.1016/S0012-365X(97)00203-3; F.M. Dong et al., “Sharp bounds for the number of 3-independent partitions and the chromaticity of bipartite graphs,” J. Graph Theory 37 (2001), 48–77; and R. Hasni/Y.H. Peng, “Chromatic Uniqueness of Complete Bipartite Graphs with Certain Edges Deleted II,” Punjab Univ. J. Math. 43 (2011), 1–8.

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