Hamilton decompositions of regular expanders: applications

TL;DR

This paper extends the theory of Hamilton decompositions in regular expanders, deriving new results for undirected graphs, dense graphs, and random graphs, confirming several longstanding conjectures.

Contribution

It generalizes Hamilton decomposition results to undirected graphs, dense graphs, and random graphs, and verifies multiple conjectures in these contexts.

Findings

Undirected analogue of robust outexpander result

Optimal bounds on Hamilton cycle packings in graphs with minimum degree d

Verification of conjectures in random graphs and tournaments

Abstract

In a recent paper, we showed that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. The main consequence of this theorem is that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large. This verified a conjecture of Kelly from 1968. In this paper, we derive a number of further consequences of our result on robust outexpanders, the main ones are the following: (i) an undirected analogue of our result on robust outexpanders; (ii) best possible bounds on the size of an optimal packing of edge-disjoint Hamilton cycles in a graph of minimum degree d for a large range of values for d. (iii) a similar result for digraphs of given minimum semidegree; (iv) an approximate version of a conjecture of Nash-Williams on Hamilton decompositions of dense regular graphs; (v) the observation that dense quasi-random graphs are robust outexpanders; (vi) a verification of the `very dense' case of a conjecture of Frieze and Krivelevich on packing edge-disjoint Hamilton cycles in random graphs; (vii) a proof of a conjecture of Erdos on the size of an optimal packing of edge-disjoint Hamilton cycles in a random tournament.