Abstract. The canonical Ramsey theorem of Erdős and Rado implies that for any graph [Formula: see text], any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph [Formula: see text] contains a monochromatic, lexicographic, or rainbow copy of [Formula: see text]. The least such [Formula: see text] is called the Erdős–Rado number of [Formula: see text], denoted by [Formula: see text]. Erdős–Rado numbers of cliques have received considerable attention, and in this paper we extend this line of research by studying Erdős–Rado numbers of sparse graphs. For example, we prove that if [Formula: see text] has bounded degree, then [Formula: see text] is polynomial in [Formula: see text] if [Formula: see text] is bipartite but exponential in general. We also study the closely related problem of constrained Ramsey numbers. For a given tree [Formula: see text] and given path [Formula: see text], we study the minimum [Formula: see text] such that every edge-coloring of [Formula: see text] contains a monochromatic copy of [Formula: see text] or a rainbow copy of [Formula: see text]. We prove a nearly optimal upper bound for this problem, which differs from the best known lower bound by a function of inverse Ackermann type.

Abstract. In this paper we prove several results on Ramsey numbers [Formula: see text] for a fixed graph [Formula: see text] and a large graph [Formula: see text], in particular for [Formula: see text]. These results extend earlier work of Erdős, Faudree, Rousseau, and Schelp and of Balister, Schelp, and Simonovits on so-called Ramsey size-linear graphs. Among other results, we show that if [Formula: see text] is a subdivision of [Formula: see text] with at least six vertices, then [Formula: see text] for every graph [Formula: see text]. We also conjecture that if [Formula: see text] is a connected graph with [Formula: see text], then [Formula: see text]. The case [Formula: see text] was proved by Erdős, Faudree, Rousseau, and Schelp. We prove the case [Formula: see text].

Abstract. Graph modification problems ask for the minimal number of vertex/edge additions/deletions needed to make a graph satisfy some predetermined property. A (meta-)problem of this type, which was raised by Yannakakis in 1981, asks to determine for which properties [Formula: see text] it is NP-hard to compute the smallest number of edge deletions needed to make a graph satisfy [Formula: see text]. Despite being extensively studied in the past 40 years, this problem is still wide open. In fact, it is open even when [Formula: see text] is the property of being [Formula: see text]-free, for some fixed graph [Formula: see text]. In this case we use [Formula: see text] to denote the smallest number of edge deletions needed to turn [Formula: see text] into an [Formula: see text]-free graph. Alon, Shapira, and Sudakov proved that if [Formula: see text] is not bipartite, then computing [Formula: see text] is NP-hard. They left open the problem of classifying the bipartite graphs [Formula: see text] for which computing [Formula: see text] is NP-hard. In this paper we resolve this problem when [Formula: see text] is a forest, showing that computing [Formula: see text] is polynomial-time solvable if [Formula: see text] is a star forest and NP-hard otherwise. Our main innovation in this work lies in introducing a new graph-theoretic approach for Yannakakis’s problem, which differs significantly from all prior works on this subject. In particular, we prove new results concerning an old and famous conjecture of Erdős and Sós, which are of independent interest.

A graph G on n vertices is ε-far from property P if one should add/delete at least ε n² edges to turn G into a graph satisfying P. A distance estimator for P is an algorithm that given G and α, ε > 0 distinguishes between the case that G is (α-ε)-close to 𝒫 and the case that G is α-far from 𝒫. If P has a distance estimator whose query complexity depends only on ε, then P is said to be estimable. Every estimable property is clearly also testable, since testing corresponds to estimating with α = ε. A central result in the area of property testing is the Fischer-Newman theorem, stating that an inverse statement also holds, that is, that every testable property is in fact estimable. The proof of Fischer and Newmann was highly ineffective, since it incurred a tower-type loss when transforming a testing algorithm for P into a distance estimator. This raised the natural problem, studied recently by Fiat-Ron and by Hoppen-Kohayakawa-Lang-Lefmann-Stagni, whether one can find a transformation with a polynomial loss. We obtain the following results. - We show that if P is hereditary, then one can turn a tester for P into a distance estimator with an exponential loss. This is an exponential improvement over the result of Hoppen et. al., who obtained a transformation with a double exponential loss. - We show that for every P, one can turn a testing algorithm for P into a distance estimator with a double exponential loss. This improves over the transformation of Fischer-Newman that incurred a tower-type loss. Our main conceptual contribution in this work is that we manage to turn the approach of Fischer-Newman, which was inherently ineffective, into an efficient one. On the technical level, our main contribution is in establishing certain properties of Frieze-Kannan Weak Regular partitions that are of independent interest.

Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e.g., planar graphs). This line of work, which started in the early 80’s, culminated in a recent work of Gishboliner et al., which highlighted the importance of the task of counting homomorphic copies of cycles (i.e., cyclic walks) in graphs of bounded degeneracy. Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of detecting (standard) copies of directed cycles in general directed graphs. More precisely, we prove the following: One can compute the number of homomorphic copies of C 2k and C 2k+1 in n -vertex graphs of bounded degeneracy in time Õ( n d k ), where the fastest known algorithm for detecting directed copies of C k in general m -edge digraphs runs in time Õ( m d k ). Conversely, one can transform any O(n b k ) algorithm for computing the number of homomorphic copies of C 2k or of C 2k+1 in n -vertex graphs of bounded degeneracy, into an Õ( m b k ) time algorithm for detecting directed copies of C k in general m -edge digraphs. We emphasize that our first result does not use a black-box reduction (as opposed to the second result which does). Instead, we design an algorithm for computing the number of C k -homomorphisms in degenerate graphs and show that one part of its analysis can be reduced to the analysis of the fastest known algorithm for detecting directed cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams. As a by-product of our algorithm, we obtain a new algorithm for detecting k -cycles in directed and undirected graphs of bounded degeneracy that is faster than all previously known algorithms for 7 ≤ k ≤ 11, and faster for all k ≥ 7 if the matrix multiplication exponent is 2.

Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e.g., planar graphs). This line of work, which started in the early 80's, culminated in a recent work of Gishboliner et al., which highlighted the importance of the task of counting homomorphic copies of cycles (i.e., cyclic walks) in graphs of bounded degeneracy. Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of detecting (standard) copies of directed cycles in general directed graphs. More precisely, we prove the following: One can compute the number of homomorphic copies of C2k and C2k+1 in n-vertex graphs of bounded degeneracy in time , where the fastest known algorithm for detecting directed copies of Ck in general m-edge digraphs runs in time . Conversely, one can transform any algorithm for computing the number of homomorphic copies of C2k or of C2k+1 in n-vertex graphs of bounded degeneracy, into an time algorithm for detecting directed copies of Ck in general m-edge digraphs. We emphasize that our first result does not use a black-box reduction (as opposed to the second result which does). Instead, we design an algorithm for computing the number of Ck-homomorphisms in degenerate graphs and show that one part of its analysis can be reduced to the analysis of the fastest known algorithm for detecting directed cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams. As a by-product of our algorithm, we obtain a new algorithm for detecting k-cycles in directed and undirected graphs of bounded degeneracy that is faster than all previously known algorithms for 7 ≤ k ≤ 11, and faster for all k ≥ 7 if the matrix multiplication exponent is 2.

We consider the problem of counting the number of copies of a fixed graph H within an input graph G . This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input G has bounded degeneracy . This is a rich family of graphs, containing all graphs without a fixed minor (e.g., planar graphs), as well as graphs generated by various random processes (e.g., preferential attachment graphs). We say that H is easy if there is a linear-time algorithm for counting the number of copies of H in an input G of bounded degeneracy. A seminal result of Chiba and Nishizeki from ’85 states that every H on at most 4 vertices is easy. Bera, Pashanasangi, and Seshadhri recently extended this to all H on 5 vertices and further proved that for every \( k \gt 5 \) there is a k -vertex H which is not easy. They left open the natural problem of characterizing all easy graphs H . Bressan has recently introduced a framework for counting subgraphs in degenerate graphs, from which one can extract a sufficient condition for a graph H to be easy. Here, we show that this sufficient condition is also necessary, thus fully answering the Bera–Pashanasangi–Seshadhri problem. We further resolve two closely related problems; namely characterizing the graphs that are easy with respect to counting induced copies, and with respect to counting homomorphisms.

The regularity lemma of Szemeredi turned out to be the most powerful tool for studying the testability of graph properties in the dense graph model. In fact, as we argue in this paper, this lemma can be used in order to prove (essentially) all the previous results in this area. More precisely, a barrier for obtaining an efficient testing algorithm for a graph property P was having an efficient regularity lemma for graphs satisfying P. The problem is that for many natural graph properties (e.g. triangle freeness) it is known that a graph can satisfy P and still only have regular partitions of tower-type size. This means that there was no viable path for obtaining reasonable bounds on the query complexity of testing such properties. In this paper we consider the property of being induced C_4-free, which also suffers from the fact that a graph might satisfy this property but still have only regular partitions of tower-type size. By developing a new approach for this problem we manage to overcome this barrier and thus obtain a merely exponential bound for testing this property. This is the first substantial progress on a problem raised by Alon in 2001, and more recently by Alon, Conlon and Fox. We thus obtain the first example of an efficient testing algorithm that cannot be derived from an efficient version of the regularity lemma.

Our first theorem in this paper is a hierarchy theorem for the query complexity of testing graph properties with 1-sided error; more precisely, we show that for every sufficiently fast-growing function f, there is a graph property whose 1-sided-error query complexity is precisely f(Θ(1/ε)). No result of this type was previously known for any f which is super-polynomial. Goldreich [ECCC 2005] asked to exhibit a graph property whose query complexity is 2Θ(1/ε). Our hierarchy theorem partially resolves this problem by exhibiting a property whose 1-sided-error query complexity is 2Θ(1/ε). We also use our hierarchy theorem in order to resolve a problem raised by the second author and Alon [STOC 2005] regarding testing relaxed versions of bipartiteness.

We give new sufficient and necessary criteria guaranteeing that