Nash-Williams's Hamilton cycle packing conjecture

Let $G$ be a graph on $n$ vertices, let $\delta(G)$ denote its minimum degree, and let $\operatorname{reg}_{\operatorname{even}}(G)$ denote the degree of a largest even-regular spanning subgraph of $G$.

Nash-Williams's conjecture. Suppose that $G$ is a graph on $n$ vertices with minimum degree

$$\delta(G)\ge n/2.$$

Then $G$ contains $\operatorname{reg}_{\operatorname{even}}(G)/2$ edge-disjoint Hamilton cycles.

This conjecture would simultaneously generalize the Hamilton decomposition and Hamilton cycle-packing results discussed in the paper, and would give a best-possible bound for each graph rather than only for graphs with a prescribed minimum-degree class. The source records that it was proved for $\delta\ge(2-\sqrt{2}+\varepsilon)n$, so the full statement remains open in the supplied context.