On covering expander graphs by Hamilton cycles

TL;DR

This paper demonstrates that graphs with certain expansion properties and many edge-disjoint Hamilton cycles can be nearly perfectly covered by Hamilton cycles, extending to random graphs and nearly matching optimal bounds.

Contribution

It proves that such graphs can be covered by slightly more than half the maximum degree in Hamilton cycles, generalizing previous results and applying to random graphs.

Findings

Graphs with expansion properties and many edge-disjoint Hamilton cycles can be covered by nearly the same number of Hamilton cycles.

For random graphs G(n,p), all edges can be covered by approximately np/2 Hamilton cycles asymptotically almost surely.

The results are nearly optimal, matching known bounds up to lower order terms.

Abstract

The problem of packing Hamilton cycles in random and pseudorandom graphs has been studied extensively. In this paper, we look at the dual question of covering all edges of a graph by Hamilton cycles and prove that if a graph with maximum degree $\Delta$ satisfies some basic expansion properties and contains a family of $(1-o(1))\Delta/2$ edge disjoint Hamilton cycles, then there also exists a covering of its edges by $(1+o(1))\Delta/2$ Hamilton cycles. This implies that for every $\alpha >0$ and every $p \geq n^{\alpha-1}$ there exists a covering of all edges of $G(n,p)$ by $(1+o(1))np/2$ Hamilton cycles asymptotically almost surely, which is nearly optimal.