On packing Hamilton cycles in -regular graphs

A graph G = ( V , E ) on n vertices is ( α , ε ) -regular if its minimal degree is at least α n , and for every pair of disjoint subsets S , T ⊂ V of cardinalities at least ε n , the number of edges e ( S , T ) between S and T satisfies e ( S , T ) | S | | T | - α ⩽ ε . We prove that if α ⪢ ε > 0 are not too small, then every ( α , ε ) -regular graph on n vertices contains a family of ( α / 2 - O ( ε ) ) n edge-disjoint Hamilton cycles. As a consequence we derive that for every constant 0 < p < 1 , with high probability in the random graph G ( n , p ) , almost all edges can be packed into edge-disjoint Hamilton cycles. A similar result is proven for the directed case.