Rigorous Asymptotics for Hamiltonian Cycles in Random Maps

Definitions and mathematical statement

A map is a finite connected graph cellularly embedded in a closed surface, so that each complementary face is a topological disc. A map is trivalent if every vertex has degree three, and vertex bicolored if its vertices have two colors with adjacent vertices of different colors. A Hamiltonian cycle traverses every vertex exactly once before returning to its starting vertex. Conventions concerning multiple edges, cycle orientation, and counting multiplicities must be specified for an enumeration problem.

The supplied figure caption describes a vertex-bicolored trivalent map of genus zero, with \[ F-E+V=3-3+2=2=2-2g,\qquad g=0. \] It also describes a redrawing displaying a given Hamiltonian cycle and a rooted edge indicated by scissors. These descriptions provide context but do not specify the full class of maps or the rooting convention used in the requested asymptotics.

Requested statement, with unresolved parameters. Establish by rigorous probabilistic or combinatorial methods the predicted asymptotic behavior associated with Hamiltonian cycles in the intended class of random maps. The supplied record refers to "this prediction" without supplying its formula. In particular, it does not specify the counted quantity or random variable, the probability measure, the size parameter and its admissible values, the genus restrictions, the exponential growth constant, the critical exponent, or the strength of the asymptotic assertion. Consequently, a precise quantified asymptotic conjecture cannot be recovered from this record alone. The figure's genus-zero example does not establish a genus-zero restriction for the problem.

Short English statement

The problem asks for a rigorous probabilistic or combinatorial proof of a physics prediction concerning Hamiltonian cycles in random maps. The prediction and the random-map model are missing, so the supplied material does not determine a precise asymptotic statement. The illustrated bicolored trivalent planar map supplies context requiring clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005330$, exact source problem code catalogue entry, title: Rigorous Asymptotics for Hamiltonian Cycles in Random Maps. Collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2022.
• Supplied source citation: Enumerative Combinatorics (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/57, pp. 3215--3305. https://doi.org/10.4171/owr/2022/57. The supplied extract contains a question referring to an omitted prediction and a Figure 5 caption.
• Supplied literature source labeled "Di Francesco et al.": https://arxiv.org/abs/2305.02188. No title, full author list, or specific formula is supplied.
• Supplied literature source labeled "original OWR report": https://doi.org/10.4171/owr/2022/57.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that physics analyses predict the Hamiltonian-cycle growth and critical exponent and provide numerical evidence, while a rigorous proof of the requested asymptotics remains unavailable. This is supplied dated curation, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.