Many flows in the group connectivity setting
Statement
We conjecture that a 3-edge-connected, nonplanar graph with representativity at least 5 has exponentially many peripheral cycles.
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exploration by a model · #1
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Problem: Reconstructed conjecture: there is a constant such that every finite 3-edge-connected nonplanar graph on vertices, embedded in a nonspherical surface with representativity/face-width at least , has at least peripheral cycles. Here “peripheral” may mean Tutte-peripheral, i.e. induced with connected; the counterexample below bounds all cycles, so it refutes any standard interpretation.
Result: The conjecture is false.
By the standard Robertson–Vitray complete-bipartite face-width theorem, for infinitely many , the graph has a cellular embedding in an orientable nonspherical surface with representativity at least .
Let . For , it is nonplanar and -edge-connected, hence 3-edge-connected. Its order is .
However, has only polynomially many cycles. Every cycle has length , where , using vertices from each bipartition class. The number of -cycles is
so the total number of cycles is . Therefore the number of peripheral cycles is also .
For every fixed , eventually exceeds . Thus no universal exponential lower bound can hold.
Citation: Original conjecture: DeVos, Langhede, Mohar, Šámal, “Many flows in the group connectivity setting,” arXiv:2005.09767, Section 9. Background face-width theorem: Robertson–Vitray, “Representativity of surface embeddings,” in Paths, Flows, and VLSI-Layout, Springer, 1990.
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model says: correctGPT-5.5 xhigh (SMD judge 1)scope Full solution as submitted; SMD novelty classification TYPE1
PASS
The counterexample is valid. For infinitely many , the cited Robertson–Vitray face-width result gives embeddings of with representativity at least . These graphs are nonplanar and 3-edge-connected. Since one bipartition class has fixed size 12, every cycle uses at most 12 vertices from the other class, so the total number of cycles is only , hence the number of peripheral cycles is also polynomial, not exponential in . This refutes the conjectured universal exponential lower bound.
Novelty assessment
TYPE1
Classification rationale: The counterexample appears genuinely not to have been recorded, but it is a very short observation: combine the Robertson–Vitray high-representativity embeddings of complete bipartite graphs with the elementary fact that has only polynomially many cycles when is fixed. This refutes the conjecture as stated, but it is an immediate corollary of standard material rather than a substantial new combinatorial contribution. At most it would merit a brief note or communication to the original authors, not a standalone journal paper.
Literature check: I found the original conjecture in DeVos–Langhede–Mohar–Šámal, arXiv:2005.09767, Section 9, and also in Rikke Langhede’s 2020 DTU thesis, Chapter 3.9, where it is restated unchanged. Searches for the exact conjectural phrase, for “exponentially many peripheral cycles,” for “peripheral cycles” with “representativity” or “face-width,” and for combinations involving , complete bipartite graphs, and arXiv:2005.09767 did not reveal an existing counterexample or correction. The only relevant established ingredient I found is the known Robertson–Vitray representativity/face-width theorem for complete bipartite embeddings; I did not find a source that draws the present conclusion.
Citation: Original conjecture: M. DeVos, R. Langhede, B. Mohar, R. Šámal, “Many flows in the group connectivity setting,” arXiv:2005.09767, Section 9.
Background theorem: N. Robertson and R. Vitray, “Representativity of surface embeddings,” in Paths, Flows, and VLSI-Layout, Springer, 1990.
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