The effect of two cycles on the complexity of colouring by directed graphs
Discrete Applied Mathematics
Discrete Mathematics - Algebraic graph theory; a volume dedicated to Gert Sabidussi
The algorithmic aspects of the regularity lemma
Journal of Algorithms
Random sampling and approximation of MAX-CSP problems
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Robust Characterizations of Polynomials withApplications to Program Testing
SIAM Journal on Computing
SIAM Journal on Discrete Mathematics
A Lower Bound for Testing 3-Colorability in Bounded-Degree Graphs
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Testing subgraphs in directed graphs
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Regular Languages are Testable with a Constant Number of Queries
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Property testing and its connection to learning and approximation
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Testing Subgraphs in Large Graphs
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Three theorems regarding testing graph properties
Random Structures & Algorithms
Turán Numbers of Bipartite Graphs and Related Ramsey-Type Questions
Combinatorics, Probability and Computing
A characterization of easily testable induced subgraphs
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
A Characterization of the (natural) Graph Properties Testable with One-Sided Error
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Testing triangle-freeness in general graphs
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Property Testing: A Learning Theory Perspective
Foundations and Trends® in Machine Learning
A dirac-type result on hamilton cycles in oriented graphs
Combinatorics, Probability and Computing
Note: A combinatorial proof of the Removal Lemma for Groups
Journal of Combinatorial Theory Series A
Green's conjecture and testing linear-invariant properties
Proceedings of the forty-first annual ACM symposium on Theory of computing
Triangle packings and 1-factors in oriented graphs
Journal of Combinatorial Theory Series B
Algorithmic and Analysis Techniques in Property Testing
Foundations and Trends® in Theoretical Computer Science
Directed graphs without short cycles
Combinatorics, Probability and Computing
Cycles of given length in oriented graphs
Journal of Combinatorial Theory Series B
Hamiltonian degree sequences in digraphs
Journal of Combinatorial Theory Series B
Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses
Proceedings of the forty-second ACM symposium on Theory of computing
Lower bounds for testing triangle-freeness in Boolean functions
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
A Semiexact Degree Condition for Hamilton Cycles in Digraphs
SIAM Journal on Discrete Mathematics
Introduction to testing graph properties
Property testing
Green's conjecture and testing linear invariant properties
Property testing
Introduction to testing graph properties
Property testing
Green's conjecture and testing linear invariant properties
Property testing
Introduction to testing graph properties
Studies in complexity and cryptography
An approximate version of Sumner's universal tournament conjecture
Journal of Combinatorial Theory Series B
Testing Eulerianity and connectivity in directed sparse graphs
Theoretical Computer Science
Subgraphs of Weakly Quasi-Random Oriented Graphs
SIAM Journal on Discrete Mathematics
A survey on Hamilton cycles in directed graphs
European Journal of Combinatorics
On the query complexity of testing orientations for being Eulerian
ACM Transactions on Algorithms (TALG)
Nearly complete graphs decomposable into large induced matchings and their applications
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
SIAM Journal on Discrete Mathematics
Embedding cycles of given length in oriented graphs
European Journal of Combinatorics
Hamilton decompositions of regular expanders: Applications
Journal of Combinatorial Theory Series B
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Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least εn2 edges have to be deleted from it to make it H-free. We show that in this case G contains at least f(ε, H)nh copies of H. This is proved by establishing a directed version of Szemerédi's regularity lemma, and implies that for every H there is a one-sided error property tester whose query complexity is bounded by a function of ε only for testing the property PH of being H-free.As is common with applications of the undirected regularity lemma, here too the function 1/f(ε,H) is an extremely fast growing function in ε. We therefore further prove a precise characterization of all the digraphs H, for which f(ε,H) has a polynomial dependency on ε. This implies a characterization of all the digraphs H, for which the property of being H-free has a one-sided error property tester whose query complexity is polynomial in 1/ε. We further show that the same characterization also applies to two-sided error property testers as well. A special case of this result settles an open problem raised by the first author in (Alon, Proceedings of the 42nd IEEE FOCS, IEEE, New York, 2001, pp. 434-441). Interestingly, it turns out that if PH has a polynomial query complexity, then there is a two-sided ε-tester for PH that samples only O(1/ε) vertices, whereas any one-sided tester for PH makes at least (1/ε)d/2 queries, where d is the average degree of H. We also show that the complexity of deciding if for a given directed graph H, PH has a polynomial query complexity, is NP-complete, marking an interesting distinction from the case of undirected graphs.For some special cases of directed graphs H, we describe very efficient one-sided error property-testers for testing PH. As a consequence we conclude that when H is an undirected bipartite graph, we can give a one-sided error property tester with query complexity O((1/ε)h/2), improving the previously known upper bound O((1/ε)h2). The proofs combine combinatorial, graph theoretic and probabilistic arguments with results from additive number theory.