Cycle factors in randomly perturbed graphs

We study the problem of finding pairwise vertex-disjoint copies of the ω>-vertex cycle Cω>in the randomly perturbed graph model, which is the union of a deterministic n-vertex graph G and the binomial random graph G(n, p). For ω>≥ 3 we prove that asymptotically almost surely G U G(n, p) contains min{δ(G), min{δ(G), [n/l]} pairwise vertex-disjoint cycles Cω>, provided p ≥ C log n/n for C sufficiently large. Moreover, when δ(G) ≥ αn with 0 ≤ α/l and G and is not 'close' to the complete bipartite graph Kαn(1 - α)n, then p ≥ C/n suffices to get the same conclusion. This provides a stability version of our result. In particular, we conclude that p ≥ C/n suffices when α > n/l for finding [n/l] cycles Cω>. Our results are asymptotically optimal. They can be seen as an interpolation between the Johansson-Kahn-Vu Theorem for Cω>-factors and the resolution of the El-Zahar Conjecture for Cω>-factors by Abbasi

Decomposing graphs of high minimum degree into 4-cycles

If a graph G decomposes into edge-disjoint 4-cycles, then each vertex of G has even degree and 4 divides the number of edges in G. It is shown that these obvious necessary conditions are also sufficient when G is any simple graph having minimum degree at least (3132+on(1))n, where n is the number of vertices in G. This improves the bound given by Gustavsson (PhD Thesis, University of Stockholm, 1991), who showed (as part of a more general result) sufficiency for simple graphs with minimum degree at least (1-10-94+on(1))n. On the other hand, it is known that for arbitrarily large values of n there exist simple graphs satisfying the obvious necessary conditions, having n vertices and minimum degree 35n-1, but having no decomposition into edge-disjoint 4-cycles. We also show that if G is a bipartite simple graph with n vertices in each part, then the obvious necessary conditions for G to decompose into 4-cycles are sufficient when G has minimum degree at least (3132+on(1))n

Vertex covers by monochromatic pieces - A survey of results and problems

This survey is devoted to problems and results concerning covering the vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles and other objects. It is an expanded version of the talk with the same title at the Seventh Cracow Conference on Graph Theory, held in Rytro in September 14-19, 2014.Comment: Discrete Mathematics, 201

Hamilton cycles, minimum degree and bipartite holes

We present a tight extremal threshold for the existence of Hamilton cycles in graphs with large minimum degree and without a large ``bipartite hole`` (two disjoint sets of vertices with no edges between them). This result extends Dirac's classical theorem, and is related to a theorem of Chv\'atal and Erd\H{o}s. In detail, an $(s, t)$-bipartite-hole in a graph $G$ consists of two disjoint sets of vertices $S$ and $T$ with $|S|= s$ and $|T|=t$ such that there are no edges between $S$ and $T$; and $\widetilde{\alpha}(G)$ is the maximum integer $r$ such that $G$ contains an $(s, t)$-bipartite-hole for every pair of non-negative integers $s$ and $t$ with $s + t = r$. Our central theorem is that a graph $G$ with at least $3$ vertices is Hamiltonian if its minimum degree is at least $\widetilde{\alpha}(G)$. From the proof we obtain a polynomial time algorithm that either finds a Hamilton cycle or a large bipartite hole. The theorem also yields a condition for the existence of $k$ edge-disjoint Hamilton cycles. We see that for dense random graphs $G(n,p)$, the probability of failing to contain many edge-disjoint Hamilton cycles is $(1 - p)^{(1 + o(1))n}$. Finally, we discuss the complexity of calculating and approximating $\widetilde{\alpha}(G)$