Hamiltonicity of a uniform fixed-degree regular graph

For each fixed integer $d\ge 3$, the probability that a uniformly sampled labelled simple $d$-regular graph is Hamiltonian tends to one as its order grows through feasible values. A Hamiltonian cycle is a spanning cyclic ordering with an edge between every consecutive pair, including the last and first. The result is asymptotic and imposes no chosen Hamiltonian cycle in advance.

Theorem 1 of Robinson and Wormald is imported under the uniform labelled model and fixed-degree/parity conventions stated immediately above it on page 2. No local proof of this substantial input is claimed. In particular, the perfect-matching concentration estimate (2.1) is not inferred merely from the pairing construction or from the first moment of a cycle count.

Pairings, simple-graph conditioning and model transfer defines the uniform simple regular graph law and distinguishes it from the unconditioned pairing multigraph.

Finite symmetric networks and loop removal fixes the simple graph and network conventions; the theorem concerns the simple unweighted case.

Asymptotic quantifiers and uniform parameter ranges defines the feasible-order high-probability limit; the degree remains fixed.

Feasible orders for labelled simple regular graphs establishes the exact feasible orders.

Hamiltonian cycles as finite cyclic-order events establishes the finite Hamilton-cycle event and cycle convention.

Vertex relabeling preserves the uniform regular graph law establishes the relabeling action preserving the uniform law.

Intersecting finitely many high-probability events establishes simultaneous high-probability intersections.

From Hamiltonicity and an invariant event to a prescribed cycle establishes the conditional existence transfer to any prescribed cycle.

For any relabeling-invariant properties $Q_n$ whose probabilities tend to one in this same fixed-degree model, there is, for every sufficiently large feasible $n$, a graph in $Q_n$ containing any prescribed Hamiltonian cycle on $[n]$. Apply the prescribed-cycle transfer lemma to the imported Hamiltonicity limit and the assumed limit for $Q_n$. This local consequence does not assert that a uniform graph contains a preselected cycle with high probability; invariance of $Q_n$ is essential to the relabeling argument.