Tilings in randomly perturbed dense graphs

A perfect $H$-tiling in a graph $G$ is a collection of vertex-disjoint copies of a graph $H$ in $G$ that together cover all the vertices in $G$. In this paper we investigate perfect $H$-tilings in a random graph model introduced by Bohman, Frieze and Martin in which one starts with a dense graph and then adds $m$ random edges to it. Specifically, for any fixed graph $H$, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect $H$-tiling with high probability. Our proof utilises Szemer\'edi's Regularity lemma as well as a special case of a result of Koml\'os concerning almost perfect $H$-tilings in dense graphs.Comment: 19 pages, to appear in CP

Rainbow connectivity of randomly perturbed graphs

In this note we examine the following random graph model: for an arbitrary graph $H$, with quadratic many edges, construct a graph $G$ by randomly adding $m$ edges to $H$ and randomly coloring the edges of $G$ with $r$ colors. We show that for $m$ a large enough constant and $r \geq 5$, every pair of vertices in $G$ are joined by a rainbow path, i.e., $G$ is {\it rainbow connected}, with high probability. This confirms a conjecture of Anastos and Frieze [{\it J. Graph Theory} {\bf 92} (2019)] who proved the statement for $r \geq 7$ and resolved the case when $r \leq 4$ and $m$ is a function of $n$.Comment: Some typos and errors fixe

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

Smoothed Analysis of the Koml\'os Conjecture: Rademacher Noise

The {\em discrepancy} of a matrix $M \in \mathbb{R}^{d \times n}$ is given by $\mathrm{DISC}(M) := \min_{\boldsymbol{x} \in \{-1,1\}^n} \|M\boldsymbol{x}\|_\infty$. An outstanding conjecture, attributed to Koml\'os, stipulates that $\mathrm{DISC}(M) = O(1)$, whenever $M$ is a Koml\'os matrix, that is, whenever every column of $M$ lies within the unit sphere. Our main result asserts that $\mathrm{DISC}(M + R/\sqrt{d}) = O(d^{-1/2})$ holds asymptotically almost surely, whenever $M \in \mathbb{R}^{d \times n}$ is Koml\'os, $R \in \mathbb{R}^{d \times n}$ is a Rademacher random matrix, $d = \omega(1)$, and $n = \tilde \omega(d^{5/4})$. We conjecture that $n = \omega(d \log d)$ suffices for the same assertion to hold. The factor $d^{-1/2}$ normalising $R$ is essentially best possible.Comment: For version 2, the bound on the discrepancy is improve

Karp's patching algorithm on random perturbations of dense digraphs

We consider the following question. We are given a dense digraph $D_0$ with minimum in- and out-degree at least $\alpha n$, where $\alpha>0$ is a constant. We then add random edges $R$ to $D_0$ to create a digraph $D$. Here an edge $e$ is placed independently into $R$ with probability $n^{-\epsilon}$ where $\epsilon>0$ is a small positive constant. The edges of $D$ are given edge costs $C(e),e\in E(D)$, where $C(e)$ is an independent copy of the exponential mean one random variable $EXP(1)$ i.e. $\Pr(EXP(1)\geq x)=e^{-x}$. Let $C(i,j),i,j\in[n]$ be the associated $n\times n$ cost matrix where $C(i,j)=\infty$ if $(i,j)\notin E(D)$. We show that w.h.p. the patching algorithm of Karp finds a tour for the asymmetric traveling salesperson problem that is asymptotically equal to that of the associated assignment problem. Karp's algorithm runs in polynomial time.Comment: Fixed the proof of a lemm