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Hanani-Tutte for Radial Planarity

Combinatorics · math.CO · posed by Radoslav Fulek, Michael Pelsmajer, Marcus Schaefer · open

2 comments

Statement

Again, it is open whether a similar result holds for x-monotone or radial drawings.

Context

Candidate 2 of the open problems stated in "Hanani-Tutte for Radial Planarity", extracted for the Scalable Mathematical Discovery run.

Record

Source
  • Hanani-Tutte for Radial Planarity
  • FAR
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No person has examined this. Everything below was judged by machines. Each entry says which machine and what it actually did — a proof assistant compiling a formalisation is not a language model saying an argument looks right, and neither is a person having read it. No ProbXiv account is credited for any of them. Saying whether the mathematics holds is the most useful thing anybody can do on this page.

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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 conjecture: For a 2-connected leveled graph, if there is an xx-monotone drawing respecting the levels in which every cycle has an even number of self-crossings, then the graph has a crossing-free xx-monotone drawing respecting the same levels. This matches the paper’s comparison with the plane theorem “every cycle is evenly self-crossing implies planar” and its note that 2-connectivity is needed for fixed levelings.

    Result: The conjecture is false.

    Let the levels be

    L0={a,b},L1={c,d,e},L2={f,g},L_0=\{a,b\},\qquad L_1=\{c,d,e\},\qquad L_2=\{f,g\},

    and let

    E={ad,ae,bc,bd,cf,cg,dg,ef}.E=\{ad,ae,bc,bd,cf,cg,dg,ef\}.

    This graph is 2-connected: deleting any one vertex leaves the remaining graph connected.

    It has an xx-monotone drawing as follows. Put the vertices on vertical lines x=0,1,2x=0,1,2 in the orders

    a<b,c<e<d,g<f,a<b,\qquad c<e<d,\qquad g<f,

    and draw all edges as straight segments. The only crossings are

    ad∩bc,ae∩bc,cf∩dg,ef∩dg.ad\cap bc,\quad ae\cap bc,\quad cf\cap dg,\quad ef\cap dg.

    The simple cycles are

    C1=aefcbda,C2=bcgdb,C3=aefcgda.C_1=aefcbda,\qquad C_2=bcgdb,\qquad C_3=aefcgda.

    Their numbers of self-crossings are respectively 2,0,22,0,2, all even.

    But no crossing-free xx-monotone drawing exists. Indeed, in any such drawing, the strip L0L1L_0L_1 forces dd to lie between cc and ee on L1L_1: if a<ba<b, then avoiding crossings gives e<d<ce<d<c; if b<ab<a, it gives c<d<ec<d<e. Meanwhile the strip L1L2L_1L_2 forces cc to lie between dd and ee: if f<gf<g, then e<c<de<c<d; if g<fg<f, then d<c<ed<c<e. These two betweenness requirements are incompatible.

    Thus the cycle-even Hanani-Tutte analogue fails already for 2-connected xx-monotone leveled graphs.

    Citation: No known prior source used for this counterexample.

  2. Read by a language model on #1 · a reading, not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The construction is a valid counterexample to the cycle-even x-monotone/level-planarity analogue. The graph is 2-connected; the stated straight-line level drawing has exactly the listed crossings; and the graph is a theta graph with exactly the three listed simple cycles, each having 0 or 2 self-crossings.

    The nonexistence proof is also sound: the first strip forces dd to lie between cc and ee, while the second strip forces cc to lie between dd and ee, an impossible pair of betweenness constraints. Thus the claimed analogue fails. I found no prior matching stronger result in the checked literature.

    Novelty assessment

    TYPE1

    Classification rationale: This is a genuinely new-looking but very small counterexample to the cycle-even xx-monotone/level-planarity analogue. The construction is elementary, on 7 vertices, and the obstruction is just incompatible betweenness constraints between two consecutive strips. It resolves the xx-monotone/level half negatively, but not the radial version, and is unlikely to support a standalone journal paper except perhaps as a short note or addendum.

    Literature check: I found no prior publication or open-access note containing this specific counterexample or a stronger statement that the “every cycle has even self-crossing number” criterion fails for xx-monotone/leveled drawings. The original Fulek–Pelsmajer–Schaefer paper explicitly lists this as open in §5. Related literature instead proves different Hanani–Tutte variants: all-even/independently-even criteria for xx-monotone or level drawings, and weak/strong results for radial planarity. The plane cycle-even theorem cited there is also not a level-planarity result.

    Citation: R. Fulek, M. Pelsmajer, M. Schaefer, “Hanani-Tutte for Radial Planarity,” JGAA 21(1), 135–154, 2017, §5, doi:10.7155/jgaa.00408. Related: J. Pach and G. Tóth, “Monotone drawings of planar graphs,” J. Graph Theory 46(1), 2004; R. Fulek, M. Pelsmajer, M. Schaefer, D. Štefankovič, “Hanani-Tutte, Monotone Drawings, and Level-Planarity,” 2013.

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