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We present a linear-time algorithm for deciding first-order logic (FOL) properties in classes of graphs with bounded expansion. Many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs of bounded degree, graphs with no sub graph isomorphic to a subdivision of a fixed graph, and(More)
Green [Geometric and Functional Analysis 15 (2005), 340–376] established a version of the Szemerédi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the(More)
Wang and Lih conjectured that for every g ≥ 5, there exists a number M(g) such that the square of a planar graph G of girth at least g and maximum degree ∆ ≥ M(g) is (∆+1)-colorable. The conjecture is known to be true for g ≥ 7 but false for g ∈ {5, 6}. We show that the conjecture for g = 6 is off by just one, i.e., the square of a planar graph G of girth(More)
We prove a removal lemma for systems of linear equations over finite fields: let X1, . . . , Xm be subsets of the finite field Fq and let A be a (k×m) matrix with coefficients in Fq and rank k; if the linear system Ax = b has o(q) solutions with xi ∈ Xi, then we can destroy all these solutions by deleting o(q) elements from each Xi. This extends a result of(More)
We investigate, using purely combinatorial methods, structural and algorithmic properties of linear equivalence classes of divisors on tropical curves. In particular, an elementary proof of the RiemannRoch theorem for tropical curves, similar to the recent proof of the Riemann-Roch theorem for graphs by Baker and Norine, is presented. In addition, a(More)
A CNF formula &psi; is <i>k</i>-satisfiable if each <i>k</i> clauses of &psi; can be satisfied simultaneously. Let &pi;<inf><i>k</i></inf> be the largest real number such that for each <i>k</i>-satisfiable formula with variables <i>x</i><inf><i>i</i></inf>, there are probabilities <i>p</i><inf><i>i</i></inf> with the following property: If each variable(More)