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- Zdenek Dvorak, Daniel Král, Robin Thomas
- 2010 IEEE 51st Annual Symposium on Foundations of…
- 2010

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)

- Tomás Kaiser, Daniel Král, Serguei Norine
- Electronic Notes in Discrete Mathematics
- 2005

We show that any cubic bridgeless graph with m edges contains two perfect matchings that cover at least 3m/5 edges, and three perfect matchings that cover at least 27m/35 edges.

- Daniel Král, Oriol Serra, Lluís Vena
- J. Comb. Theory, Ser. A
- 2009

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)

- Daniel Král, Ladislav Stacho
- Journal of Graph Theory
- 2010

We show that every plane graph with maximum face size four whose all faces of size four are vertex-disjoint is cyclically 5-colorable. This answers a question of Albertson whether graphs drawn in the plane with all crossings independent are 5-colorable.

- Daniel Král, Riste Skrekovski
- SIAM J. Discrete Math.
- 2003

- Zdenek Dvorak, Daniel Král, Pavel Nejedlý, Riste Skrekovski
- Eur. J. Comb.
- 2008

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)

- Jan Hladký, Daniel Král, Serguei Norine
- J. Comb. Theory, Ser. A
- 2013

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)

- Daniel Král
- SODA
- 2003

A CNF formula ψ is <i>k</i>-satisfiable if each <i>k</i> clauses of ψ can be satisfied simultaneously. Let π<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)