arXiv subjects

SangHyun Park

Publications and source records attributed to SangHyun Park.

7 recordsLinked to original sources

arXiv2026

Hamilton decompositions of equal-side directed tori

Let $D_d(m)$ be the Cartesian product of $d$ positively oriented directed cycles of length $m$. We prove that $D_d(m)$ decomposes into $d$ directed Hamilton cycles for every $m\ge3$ and $d\ge2$. The construction proceeds by splitting coordinate directions in directed multitori. An integer selection theorem supplies unit voltages compatible with the prescribed arc multiplicities; at even modulus, the decisive condition is the parity of each incidence component. For even $m$ and odd $d\ge7$, we satisfy this condition with a factorization having one additional cycle. A relative lifting theorem preserves the first-return data of a three-colour recolouring through successive coordinate splits, after which the recolouring removes the additional cycle on a set whose size is independent of the dimension. Explicit constructions complete the low-dimensional cases. The theorem yields Hamilton decompositions of Cartesian products of equal-order, equal-degree Hamilton-decomposable digraphs, of abelian Cayley digraphs whose generators partition into module bases, and of a family of height-stretched directed tori.

math.CO↗

arXiv2026

Short Second Proof of the Odd-Modulus Directed Torus Hamilton Decomposition Theorem

Let $D_d(m)=\operatorname{Cay}((\mathbb Z/m\mathbb Z)^d,\{e_1,\ldots,e_d\})$, with all generators oriented positively. We give a second proof that $D_d(m)$ decomposes into $d$ directed Hamilton cycles for every $d\ge 2$ and every odd $m\ge 3$. The combinatorial core is a fixed-row-sum selection theorem for replicated supports: when each indexed support $A$ is repeated in $m$ identical rows, one can select $\lfloor |A|/2\rfloor$ entries from each row so that every column total is a unit modulo $m$. Applied to the Hamilton factors using a chosen coordinate direction, these selections prescribe the voltages in a cyclic lift that splits the direction into two. In fibre coordinates, the lifted successor is $\widehat h_j(x,z)=(h_j(x),z+\mathbf 1_{\{j\in M(x)\}})$. After one traversal of the base Hamilton cycle, the fibre return is translation by the total carry. Since this carry is a unit modulo $m$, the return is a single $m$-cycle and the lifted factor is Hamilton. The new fibres also preserve the direction-constant block structure required for the next split. Iterating from a directed $m$-cycle with $d$ parallel copies of each arc yields the desired decomposition. The proof strategy was proposed with the assistance of OpenAI GPT-5.5 Pro and formally verified in Lean 4.

math.CO↗

arXiv2026

Effective Angular Asymptotics and the Sharp $D^{-3}$ Horoconvex Gap Scale

We prove first-band large-diameter asymptotics for the Dirichlet spectrum on horoconvex domains in real hyperbolic space. After Chebyshev centering, a divergent sequence compactifies to a horospherical support-envelope deficit \[V\] on \[\mathbb S^{n-1}\]. For graph domains \[r<R-V(θ)\], the first band satisfies \[ λ_{j+1}=α^2+\frac{π^2}{R^2} +\frac{2π^2}{R^3}\bigl(η_j(T_n+V)-b_0\bigr)+o(R^{-3}), \qquad j=0,1, \] where \[T_n\] is the nonlocal spherical operator with multiplier \[ψ(\ell+α)-ψ(α)\]. Consequently the horoconvex fundamental gap has the sharp \[D^{-3}\] polynomial scale, and the leading large-diameter constant is characterized by the compact variational formula \[ 16π^2\inf_{V\in\mathcal A_n} \bigl(η_1(T_n+V)-η_0(T_n+V)\bigr). \] Geodesic balls realize the polynomial scale, but an explicit admissible axial perturbation lowers the reduced leading-constant value at first order.

math.SP↗

arXiv2026

Two Arc-Disjoint Hamiltonian Paths in Finite Two-Generated Abelian Cayley Digraphs

We prove the finite abelian two-generator conjecture of Darijani--Miraftab--Witte Morris: every directed Cayley digraph on a finite abelian group with two distinct nonzero generators has two arc-disjoint Hamiltonian paths. The proof uses a cut-reflection theorem for Hamiltonian cut values in the family Cay(Z_k; a, a+1): if Z is the set of such values and N=k-1, then, with N-Z={N-z : z in Z}, dist(Z,N-Z)<=1. The proof uses sector-filling inequalities for primitive-ray multiplicities and an extremal graph recording pairs at minimal reflected distance. The estimate is sharp modulo parity: exact reflection occurs for odd k, while distance one occurs for even k. The second remaining cyclic family, Cay(Z_k; -a, a+1), is treated by an explicit quotient--fiber construction. We also prove the remaining three-factor case for Cartesian products of directed cycles. Together with the two-factor and at-least-four-factor theorems of Darijani--Miraftab--Witte Morris, this resolves their directed-cycle product conjecture for all numbers of factors.

math.CO↗

arXiv2026

Hamilton decompositions of the directed 7-torus at odd modulus via root-flat certificates and a prefix-count construction

We prove that the directed seven-dimensional equal-side torus D_7(m) = Cay((Z/mZ)^7, {e_0, e_1, ..., e_6}) admits a directed Hamilton decomposition for every odd integer m >= 3. The proof has two main contributions. First, we introduce the root-flat certificate: a named verification framework in which a Hamilton decomposition of D_n(m) follows from three local conditions on a single root flat -- row Latinness, layer bijectivity, and primitive return maps. This abstraction was used informally in the earlier odd D_5(m) construction; here it appears as a definition and a theorem, providing a common verification interface for prime-dimensional base cases. Second, for every odd m >= 7, we give a uniform prefix-coordinate construction: one-layer prefix maps, a symbol-count criterion, and explicit 7x7 count matrices produce all seven Hamilton factors without a finite search. The remaining moduli m = 3 and m = 5 are exactly the boundary where the prefix-count method provably cannot work; they are handled by finite root-flat certificates whose validity is checked in Lean 4. A Lean 4 formalization verifies the Cayley statement, with the symbolic branch and the finite boundary certificates checked in the same development.

math.CO↗

arXiv2026

Hamilton decompositions of the directed 5-torus for odd modulus

We prove that the directed five-dimensional torus $D_5(m) = \operatorname{Cay}((\mathbb{Z}_m)^5, \{e_0, e_1, e_2, e_3, e_4\})$ has a Hamilton decomposition for every odd integer $m \geq 3$. This is the first higher-dimensional case in which the return-map method requires a genuine zero-set selector rather than an odometer-type correction. The construction assigns the five outgoing generators by a cyclic layer schedule with one non-constant layer determined by a zero-set Latin table; an explicit finite exact-cover certificate proves that this layer is a matching. By cyclic symmetry, Hamiltonicity of all color classes reduces to a single normalized return map. For $m \geq 5$, an explicit first-return calculation on the section $p = 2$ gives one induced cycle whose excursion lengths sum to $m^4$. The remaining modulus $m = 3$ is settled by a printed finite cycle certificate. A companion Lean 4 formalization provides an independent machine verification of the Cayley statement and the finite certificates; source, audit scripts, and ancillary search code are available at https://github.com/aria1th/Torus-Hamilton-Decomposition-Program.

math.CO↗

arXiv2026

Hamilton decompositions of the directed 3-torus: a return-map and odometer view

We prove that the directed 3-torus D_3(m), or equivalently the Cartesian product of three directed m-cycles, admits a decomposition into three arc-disjoint directed Hamilton cycles for every integer m >= 3. The proof reduces Hamiltonicity to the m-step return maps on the layer section S=i+j+k=0. For odd m, five Kempe swaps of the canonical coloring produce return maps that are explicitly affine-conjugate to the standard 2-dimensional odometer. For even m, a sign-product invariant rules out Kempe-from-canonical constructions, and a different low-layer witness reduces after one further first-return map to a finite-defect clock-and-carry system. The remaining closure is a finite splice analysis, and the case m=4 is handled separately by a finite witness. A Lean 4 formalization accompanies the construction.

math.CO↗