Hamilton decomposition at the even-regularity bound for Dirac graphs

Prove that every sufficiently large graph G with minimum degree at least n/2 contains exactly reg_even(G)/2 edge-disjoint Hamilton cycles, where reg_even(G) is the largest even integer d such that G contains a spanning d-regular subgraph.

Background

For an n-vertex graph G, the number of edge-disjoint Hamilton cycles is bounded above by floor(delta(G)/2), while reg_even(G)/2 captures the strongest regularity-based upper bound arising from a spanning even-regular subgraph. Previous results established that graphs satisfying the Dirac minimum-degree condition contain linearly many edge-disjoint Hamilton cycles and that the conjectured quantity is attained under stronger or specialized hypotheses.

The paper explains that Csaba, Kühn, Lo, Osthus, and Treglown proved the corresponding minimum guaranteed number for graphs of prescribed minimum degree and established exact Hamilton decompositions for sufficiently large regular graphs of degree at least n/2. The general assertion for arbitrary graphs satisfying the Dirac condition remains unresolved and is identified as a conjecture in the cited literature.

References

To show that an n-vertex graph G with δ(G) ≥ n/2 contains reg_even(G)/2 edge-disjoint Hamilton cycles is still open (and was conjectured in ).

— Minimum degree edge-disjoint Hamilton cycles in random directed graphs  (2502.01631 - Ferber et al., 3 Feb 2025) in Section 1, Introduction, paragraph “Dense graphs”