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Problems in Graph Theory from Memphis

Combinatorics · math.CO · posed by Ralph J. Faudree, Cecil C. Rousseau, Richard H. Schelp · open

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Statement

Let bt(n)b_{t}(n) denote the minimum number of edges induced by any set of n / 2 vertices in the Turán graph on n vertices for KtK_{t} .If each set of ⌊n/2⌋\lfloor n/2 \rfloor vertices in a graph G of order n spans more than bt(n)b_{t}(n) edges, then G contains a KtK_{t} .

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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: Interpreting “the Turán graph for KtK_t” in the standard extremal sense as Tt−1(n)T_{t-1}(n), the conjecture says: for all finite simple graphs GG of order nn, if every ⌊n/2⌋\lfloor n/2\rfloor-vertex set spans more than bt(n)b_t(n) edges, where bt(n)b_t(n) is the minimum such number in Tt−1(n)T_{t-1}(n), then GG contains KtK_t.

    Result: The statement is false.

    Take t=3t=3 and n=10n=10. The Turán graph for K3K_3 is T2(10)=K5,5T_2(10)=K_{5,5}. A part of size 55 spans no edges, so

    b3(10)=0.b_3(10)=0.

    Let GG be the balanced blow-up of C5C_5 with each cycle vertex replaced by an independent set of size 22, and consecutive cycle classes joined completely. Then GG is triangle-free, since any triangle would project to a triangle in C5C_5.

    Also α(G)=2α(C5)=4\alpha(G)=2\alpha(C_5)=4. Hence every 55-vertex subset of GG contains an edge, i.e. spans >0=b3(10)>0=b_3(10) edges. Thus GG satisfies the hypothesis but contains no K3K_3.

    So the conjecture, as written, is disproved.

    Citation: No external citation is needed; this is the standard C5C_5-blow-up counterexample.

  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 KNOWN

    PASS

    The counterexample is valid for the conjecture as stated. For t=3,n=10t=3,n=10, the Turán graph is K5,5K_{5,5}, so b3(10)=0b_3(10)=0. The 2-fold balanced blow-up of C5C_5 is triangle-free and has independence number 44, so every 5-vertex set spans at least one edge, satisfying the hypothesis while containing no K3K_3. Thus the conjecture is rigorously disproved.

    Novelty assessment

    KNOWN

    Classification rationale: The counterexample is not a novel resolution. For t=3t=3, b3(10)=0b_3(10)=0, so any triangle-free 10-vertex graph with independence number <5<5 refutes the statement. The balanced C5C_5-blow-up is a standard extremal example in the sparse-half/local-density literature and is stronger: every half-set spans at least n2/50n^2/50 edges.

    Literature check: I found no explicit later note saying “Conjecture 14 is false,” but the exact construction is well known as the sharpness example for Erdős’s Sparse-Half Conjecture. Thus the accepted counterexample is an immediate special case of a published standard construction.

    Citation: P. Erdős, R. J. Faudree, C. C. Rousseau, R. H. Schelp, “A local density condition for triangles,” Discrete Mathematics 127 (1994), 153–161. See also W. Bedenknecht, G. O. Mota, C. Reiher, M. Schacht, “On the local density problem for graphs of given odd-girth,” J. Graph Theory 90 (2019), 137–149.

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