Sparse Pseudorandom Graphs Research Articles

Rainbow Hamilton Cycles in Randomly Colored Randomly Perturbed Dense Graphs

Given an $n$-vertex graph $H$ with minimum degree at least $d n$ for some fixed $d > 0$, the distribution $H \cup \mathbb{G}(n,p)$ over the supergraphs of $H$ is referred to as a (random) perturbation of $H$. We consider the distribution of edge-colored graphs arising from assigning each edge of the random perturbation $H \cup \mathbb{G}(n,p)$ a color, chosen independently and uniformly at random from a set of colors of size $r := r(n)$. We prove that edge-colored graphs which are generated in this manner asymptotically almost surely admit rainbow Hamilton cycles whenever the edge-density of the random perturbation satisfies $p := p(n) \geq C/n$ for some fixed $C > 0$ and $r = (1 + o(1))n$. The number of colors used is clearly asymptotically best possible. In particular, this improves on a recent result of Anastos and Frieze [J. Graph Theory, 92 (2019), pp. 405--414] in this regard. As an intermediate result, which may be of independent interest, we prove that randomly edge-colored sparse pseudorandom graphs asymptotically almost surely admit an almost spanning rainbow path.

Near-perfect clique-factors in sparse pseudorandom graphs

AbstractWe prove that, for any $t \ge 3$, there exists a constant c = c(t) > 0 such that any d-regular n-vertex graph with the second largest eigenvalue in absolute value λ satisfying $\lambda \le c{d^{t - 1}}/{n^{t - 2}}$ contains vertex-disjoint copies of kt covering all but at most ${n^{1 - 1/(8{t^4})}}$ vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo (Combinatorica24 (2004), pp. 403–426) that (n, d, λ)-graphs with n ∈ 3ℕ and $\lambda \le c{d^2}/n$ for a suitably small absolute constant c > 0 contain triangle-factors. Our arguments combine tools from linear programming with probabilistic techniques, and apply them in a certain weighted setting. We expect this method will be applicable to other problems in the field.

Finding any given 2‐factor in sparse pseudorandom graphs efficiently

AbstractGiven an ‐vertex pseudorandom graph and an ‐vertex graph with maximum degree at most two, we wish to find a copy of in , that is, an embedding so that for all . Particular instances of this problem include finding a triangle‐factor and finding a Hamilton cycle in . Here, we provide a deterministic polynomial time algorithm that finds a given in any suitably pseudorandom graph . The pseudorandom graphs we consider are ‐bijumbled graphs of minimum degree which is a constant proportion of the average degree, that is, . A ‐bijumbled graph is characterised through the discrepancy property: for any two sets of vertices and . Our condition on bijumbledness is within a log factor from being tight and provides a positive answer to a recent question of Nenadov. We combine novel variants of the absorption‐reservoir method, a powerful tool from extremal graph theory and random graphs. Our approach builds on our previous work, incorporating the work of Nenadov, together with additional ideas and simplifications.

Near-perfect clique-factors in sparse pseudorandom graphs

We prove that, for any t≥3, there exists a constant c=c(t)>0 such that any d-regular n-vertex graph with the second largest eigenvalue in absolute value λ satisfying λ≤cdt−1/nt−2 contains (1−o(1))n/t vertex-disjoint copies of Kt. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), pp. 403–426] that (n,d,λ)-graphs with n∈3N and λ≤cd2 for a suitably small absolute constant c>0 contain triangle-factors.

Powers of Hamilton cycles in pseudorandom graphs

We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph G is (e,p,k,l)-pseudorandom if for all disjoint X, Y ⊂ V(G) with |X| ≥ ep k n and |Y| ≥ ep l n we have e(X,Y) = (1±e)p|X||Y|. We prove that for all β > 0 there is an e > 0 such that an (e,p,1,2)-pseudorandom graph on n vertices with minimum degree at least βpn contains the square of a Hamilton cycle. In particular, this implies that (n,d,λ)-graphs with λ ≪ d 5/2 n − 3/2 contain the square of a Hamilton cycle, and thus a triangle factor if n is a multiple of 3. This improves on a result of Krivelevich, Sudakov and Szabo [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), no. 3, 403–426]. We also obtain results for higher powers of Hamilton cycles and establish corresponding counting versions. Our proofs are constructive, and yield deterministic polynomial time algorithms.

Extremal results for odd cycles in sparse pseudorandom graphs

We consider extremal problems for subgraphs of pseudorandom graphs. For graphs $F$ and $\Gamma$ the generalized Tur\'an density $\pi_F(\Gamma)$ denotes the density of a maximum subgraph of $\Gamma$, which contains no copy of~$F$. Extending classical Tur\'an type results for odd cycles, we show that $\pi_{F}(\Gamma)=1/2$ provided $F$ is an odd cycle and $\Gamma$ is a sufficiently pseudorandom graph. In particular, for $(n,d,\lambda)$-graphs $\Gamma$, i.e., $n$-vertex, $d$-regular graphs with all non-trivial eigenvalues in the interval $[-\lambda,\lambda]$, our result holds for odd cycles of length $\ell$, provided \[ \lambda^{\ell-2}\ll \frac{d^{\ell-1}}n\log(n)^{-(\ell-2)(\ell-3)}\,. \] Up to the polylog-factor this verifies a conjecture of Krivelevich, Lee, and Sudakov. For triangles the condition is best possible and was proven previously by Sudakov, Szab\'o, and Vu, who addressed the case when $F$ is a complete graph. A construction of Alon and Kahale (based on an earlier construction of Alon for triangle-free $(n,d,\lambda)$-graphs) shows that our assumption on $\Gamma$ is best possible up to the polylog-factor for every odd $\ell\geq 5$.

Extremal results for odd cycles in sparse pseudorandom graphs

We consider extremal problems for subgraphs of pseudorandom graphs. Our results implies that for (n,d,λ)-graphs Γ satisfyingλ2k−1≪d2kn(logn)−2(k−1)(2k−1) any subgraph G⊂Γ not containing a cycle of length 2k+1 has relative density at most 12+o(1). Up to the polylog-factor the condition on λ is best possible and was conjectured by Krivelevich, Lee and Sudakov.

Embedding graphs with bounded degree in sparse pseudorandom graphs

In this paper, we show the equivalence of somequasi-random properties for sparse graphs, that is, graphsG with edge densityp=|E(G)|/( 2 )=o(1), whereo(1)→0 asn=|V(G)|→∞. Our main result (Theorem 16) is the following embedding result. For a graphJ, writeN J(x) for the neighborhood of the vertexx inJ, and letδ(J) andΔ(J) be the minimum and the maximum degree inJ. LetH be atriangle-free graph and setd H=max{δ(J):J⊆H}. Moreover, putD H=min{2d H,Δ(H)}. LetC>1 be a fixed constant and supposep=p(n)≫n −1 D H. We show that ifG is such that Moreover, we discuss a setting under which an arbitrary graphH (not necessarily triangle-free) can be embedded inG. We also present an embedding result for directed graphs.