No abstract

In this paper we determine the local and global resilience of random graphs $G_{n, p}$ ($p \gg n^{-1}$) with respect to the property of containing a cycle of length at least $(1-\alpha)n$. Roughly speaking, given $\alpha > 0$, we determine the smallest $r_g(G, \alpha)$ with the property that almost surely every subgraph of $G = G_{n, p}$ having more than $r_g(G, \alpha) |E(G)|$ edges contains a cycle of length at least $(1 - \alpha) n$ (global resilience). We also obtain, for $\alpha < 1/2$, the smallest $r_l(G, \alpha)$ such that any $H \subseteq G$ having $\deg_H(v)$ larger than $r_l(G, \alpha) \deg_G(v)$ for all $v \in V(G)$ contains a cycle of length at least $(1 - \alpha) n$ (local resilience). The results above are in fact proved in the more general setting of pseudorandom graphs.

A set A of positive integers is a Bh-set if all the sums of the form a1 + . . . + ah, with a1,. . .,ah ∈ A and a1 ⩽ . . . ⩽ ah, are distinct. We provide asymptotic bounds for the number of Bh-sets of a given cardinality contained in the interval [n] = {1,. . .,n}. As a consequence of our results, we address a problem of Cameron and Erdős (1990) in the context of Bh-sets. We also use these results to estimate the maximum size of a Bh-sets contained in a typical (random) subset of [n] with a given cardinality.

The Frieze-Kannan regularity lemma is a powerful tool in combinatorics. It has also found applications in the design of approximation algorithms and recently in the design of fast combinatorial algorithms for boolean matrix multiplication. The algorithmic applications of this lemma require one to efficiently construct a partition satisfying the conditions of the lemma.Williams [25] recently asked if one can construct a partition satisfying the conditions of the Frieze-Kannan regularity lemma in deterministic sub-cubic time. We resolve this problem by designing anÕ(n ω ) time algorithm for constructing such a partition, where ω < 2.376 is the exponent of fast matrix multiplication. The algorithm relies on a spectral characterization of vertex partitions satisfying the properties of the Frieze-Kannan regularity lemma.

In this paper we prove that two local conditions involving the degrees and co-degrees in a graph can be used to determine whether a given vertex partition is Frieze–Kannan regular. With a more refined version of these two local conditions we provide a deterministic algorithm that obtains a Frieze–Kannan regular partition of any graphGin timeO(|V(G)|2).