Alex D. Scott

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Edwards showed that every graph of size m ≥ 1 has a bipartite subgraph of size at least m/2 + √ m/8 + 1/64− 1/8. We show that every graph of size m ≥ 1 has a bipartition in which the Edwards bound holds, and in addition each vertex class contains at most m/4 + √ m/32 + 1/256 − 1/16 edges. This is exact for complete graphs of odd order, which we show are the(More)
The class Max (r, 2)-CSP (or simply Max 2-CSP) consists of constraint satisfaction problems with at most two r-valued variables per clause. For instances with n variables and m binary clauses, we present an O(nr)-time algorithm which is the fastest polynomialspace algorithm for many problems in the class, including Max Cut. The method also proves a(More)
Szemerédi’s Regularity Lemma is an important tool for analyzing the structure of dense graphs. There are versions of the Regularity Lemma for sparse graphs, but these only apply when the graph satisfies some local density condition. In this paper, we prove a sparse Regularity Lemma that holds for all graphs. More generally, we give a Regularity Lemma that(More)
We show that a random instance of a weighted maximum constraint satisfaction problem (or max 2-csp), whose clauses are over pairs of binary variables, is solvable by a deterministic algorithm in polynomial expected time, in the “sparse” regime where the expected number of clauses is half the number of variables. In particular, a maximum cut in a random(More)
Gyárfás and Sumner independently conjectured that for every tree T and integer k there is an integer f(k, T ) such that every graph G with χ(G) > f(k, T ) contains either Kk or an induced copy of T . We prove a ‘topological’ version of the conjecture: for every tree T and integer k there is g(k, T ) such that every graph G with χ(G) > g(k, T ) contains(More)
We show that a maximum cut of a random graph below the giantcomponent threshold can be found in linear space and linear expected time by a simple algorithm. In fact, the algorithm solves a more general class of problems, namely binary 2-variable constraint satisfaction problems. In addition to Max Cut, such Max 2-CSPs encompass Max Dicut, Max 2-Lin, Max(More)