Edge-disjoint Hamilton cycles in graphs

TL;DR

This paper proves that large graphs with high minimum degree contain many edge-disjoint Hamilton cycles, nearly confirming longstanding conjectures and providing asymptotically optimal bounds.

Contribution

It offers an approximate solution to Nash-Williams' 1970 question, establishing minimum degree conditions for multiple edge-disjoint Hamilton cycles.

Findings

Graphs with minimum degree > (1/2 + α)n contain at least n/8 edge-disjoint Hamilton cycles.

Almost regular graphs with high minimum degree can be nearly decomposed into Hamilton cycles.

Provides asymptotically best possible bounds for the number of Hamilton cycles in such graphs.

Abstract

In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every \alpha > 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8 edge-disjoint Hamilton cycles. More generally, we give an asymptotically best possible answer for the number of edge-disjoint Hamilton cycles that a graph G with minimum degree \delta must have. We also prove an approximate version of another long-standing conjecture of Nash-Williams: we show that for every \alpha > 0, every (almost) regular and sufficiently large graph on n vertices with minimum degree at least $(1/2 + \alpha)n$ can be almost decomposed into edge-disjoint Hamilton cycles.