Covering cycles in sparse graphs

Abstract

Let $k \geq 2$ be an integer. Kouider and Lonc proved that the vertex set of every graph $G$ with $n \geq n_0(k)$ vertices and minimum degree at least $n/k$ can be covered by $k - 1$ cycles. Our main result states that for every $\alpha > 0$ and $p = p(n) \in (0, 1]$, the same conclusion holds for graphs $G$ with minimum degree $(1/k + \alpha)np$ that are sparse in the sense that \[ e_G(X,Y) \leq p|X||Y| + o(np\sqrt{|X||Y|}/\log^3 n) \qquad \forall X,Y\subseteq V(G). \] In particular, this allows us to determine the local resilience of random and pseudorandom graphs with respect to having a vertex cover by a fixed number of cycles. The proof uses a version of the absorbing method in sparse expander graphs.