On covering expander graphs by Hamilton cycles

Details On covering expander graphs by Hamilton cycles On covering expander graphs by Hamilton cycles

2011-11-14

arxiv.org/abs/1111.3325v1

Mathematics

Combinatorics

19 pages. arXiv admin note: some text overlap with arXiv:some math/0612751

Scientific paper

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.

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