arXiv subjects
Publications and source records attributed to Panna Tímea Fekete.
In this paper, we study the so-called log-convergence of graphs defined by Balázs Szegedy (arXiv:1504.00858). We answer his Question 4 affirmatively: the sequence of incidence graphs of projective planes over finite fields log-converges, and the limit coincides with that of a particular random graph model.
In this paper, we present a new factor of IID process based on the local algorithm introduced by Díaz, Serna, and Wormald (2007). This new approach allows us to improve the previously known upper bounds on the minimum and maximum bisection width and the maximum cut of random d-regular graphs for d > 4 by introducing a new recoloring phase after the termination of the original algorithm. As an application, we show that random 5-regular graphs asymptotically almost surely admit an internal partition, i.e., a partition of the vertex set into two nonempty classes so that every vertex has at least half of its neighbors in its own class.
Generalizing the bounded kernel results of Borgs, Chayes, Lovász, Sós and Vesztergombi (2008), we prove two Sampling Lemmas for unbounded kernels with respect to the cut norm. On the one hand, we show that given a (symmetric) kernel $U\in L^p([0,1]^2)$ for some $3 2$ here). These results are then partially extended to the case of vector valued kernels. On the other hand, we show that with high probability, the $k$-samples are also close to $U$ in the cut metric, albeit with a weaker bound of order $O((\ln k)^{-\frac12+\frac1{2p}})$ (for any appropriate $p>2$). As a corollary, we obtain that whenever $U\in L^p$ with $p>4$, the $k$-samples converge almost surely to $U$ in the cut metric as $k\to\infty$.
Alon and Shapira proved that every monotone class (closed under taking subgraphs) of undirected graphs is strongly testable, that is, under the promise that a given graph is either in the class or $\varepsilon$-far from it, there is a test using a constant number of samples (depending on $\varepsilon$ only) that rejects every graph not in the class with probability at least one half, and always accepts a graph in the class. However, their bound on the number of samples is quite large since they heavily rely on Szemerédi's regularity lemma. We study the case of posets and show that every monotone class of posets is easily testable, that is, a polynomial (of $\varepsilon^{-1}$) number of samples is sufficient. We achieve this via proving a polynomial removal lemma for posets. We give a simple classification: for every monotone class of posets, there is an $h$ such that the class is indistinguishable (every large enough poset in one class is $\varepsilon$-close to a poset in the other class) from the class of $C_h$-free posets, where $C_h$ denotes the chain with $h$ elements. This allows us to test every monotone class of posets using $O(\varepsilon^{-1})$ samples. The test has a two-sided error, but it is almost complete: the probability of refuting a poset in the class is polynomially small in the size of the poset. The analogous results hold for comparability graphs, too.
We study a variant of the color-avoiding percolation model introduced by Krause et al., namely we investigate the color-avoiding bond percolation setup on (not necessarily properly) edge-colored Erdős-Rényi random graphs. We say that two vertices are color-avoiding connected in an edge-colored graph if after the removal of the edges of any color, they are in the same component in the remaining graph. The color-avoiding connected components of an edge-colored graph are maximal sets of vertices such that any two of them are color-avoiding connected. We consider the fraction of vertices contained in color-avoiding connected components of a given size as well as the fraction of vertices contained in the giant color-avoiding connected component. Under some mild assumptions on the color-densities, we prove that these quantities converge and the limits can be expressed in terms of probabilities associated to edge-colored branching process trees. We provide explicit formulas for the limit of the normalized size of the giant color-avoiding component, and in the two-colored case we also provide explicit formulas for the limit of the fraction of vertices contained in color-avoiding connected components of a given size.