Zdenek Dvorak

Deciding First-Order Properties for Sparse Graphs

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 graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We also develop an almost linear-time algorithm for deciding FOL properties in classes of graphs with locally bounded expansion, those include classes of graphs with locally bounded tree-width or locally excluding a minor. More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a linear-time initialization the data structure allows us to test an FOL property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their addition results in a graph in the class. In addition, we design a dynamic data structure for testing existential properties or the existence of short paths between prescribed vertices in such classes of graphs. All our results also hold for relational structures and are based on the seminal result of Nesetril and Ossona de Mendez on the existence of low tree-depth colorings.

Ca-Fe3+-rich, Si-undersaturated buchite from Želénky, North-Bohemian Brown Coal Basin, Czech Republic

Si-undersaturated Ca-Fe-rich buchite produced by fossil burning of a coal seam occurs in porcelanite deposit at Želenky in the North-Bohemian Brown Coal Basin, Czech Republic. Buchite is highly inhomogeneous, dense (ρ = 2.9–3.1 g/cm 3 ), strongly magnetic rock (10 −2 −10 −1 [SI]) composed of melilite, clinopyroxene (esseneite), spinel-group minerals, hematite, Ca 2 SiO 4 , calcite, aragonite, anorthite, anhydrite, perovskite, barite, two calcium ferrites - orthorhombic Ca 2 Fe 2 3+ O 5 (srebrodolskite) and CaFe 4 3+ O 7 , barium hexaferrite - BaFe 12 3+ O 19 , along with other several poorly identified phases. Clinopyroxene is rich in esseneite component (19–63 mol. %) and in a hypothetical end-member CaFe 3+ Fe 3+ SiO 6 (up to 25 mol. %) due to Fe 3+ substitution in tetrahedral sites (up to 0.26 apfu), and also in a hypothetical component Ca(Fe 2+ Mg) 0.5 Ti 0.5 AlSiO 6 (2–36 mol. %) due to incorporation of Ti (0.012–0.18 apfu), in M1 site. Melilite shows a wide compositional variability (mol. %): gehlenite 21–93, ferrigehlenite 0–34, akermanite 1–49, ferroakermanite 0–25, soda-melilite 0–15. Spinel-group minerals show solid solutions of Mf 12–80, Mt 1–62, Sp 2–68, Hc 0–21 (mol. %). Srebrodolskite, orthorhombic Ca 2 (Fe 3+ Al) 2 O 5 is widely substituted by Si and Ti. There are negative correlations among Fe + Al vs. Si + Ti ( R 2 = 0.998), and Ca (or R 2+ ) vs. total vacancy ( R 2 = 0.94), which are consistent with an exchange vector 2R 4+ □2R 3+ -1 Ca -1 , and suggest a hypothetical end-member Ca □ (SiTi) 2 O 5 . Calcium ferrite associated with srebrodolskite, ideally CaFe 4 3+ O 7 , shows extensive compositional variability: Ca 0.6–0.8 Fe 3+ 2.7–3.6 Al 0.1–0.6 Mg 0.2–0.6 Si 0.1–0.5 O 7 . First natural occurrence of barium hexaferrite, (Ba 0.6–1.1 Ca 0.1–0.2 )(Fe 3+ 10.0–11.7 Al 0.3–0.9 Ti 0.1–0.6 Mg 0.1–0.4 Mn 0.0–0.2 ) O 19 , ideally BaFe 3+ 12 O 19 , is reported. Variations of magnetic susceptibility with temperature were also investigated. Buchite originated by fusion of Tertiary clay and residual ash with overlying Quaternary sediments (loess loam with carbonate horizon) during fossil combustion of Miocene coal seam.

Three-coloring triangle-free graphs on surfaces II. 4-critical graphs in a disk

Let G be a plane graph of girth at least five. We show that if there exists a 3-coloring phi of a cycle C of G that does not extend to a 3-coloring of G, then G has a subgraph H on O(|C|) vertices that also has no 3-coloring extending phi. This is asymptotically best possible and improves a previous bound of Thomassen. In the next paper of the series we will use this result and the attendant theory to prove a generalization to graphs on surfaces with several precolored cycles.

Graphic TSP in cubic graphs

We present a polynomial-time 9/7-approximation algorithm for the graphic TSP for cubic graphs, which improves the previously best approximation factor of 1.3 for 2-connected cubic graphs and drops the requirement of 2-connectivity at the same time. To design our algorithm, we prove that every simple 2-connected cubic n-vertex graph contains a spanning closed walk of length at most 9n/7-1, and that such a walk can be found in polynomial time.

Independent Sets near the Lower Bound in Bounded Degree Graphs

By Brooks Theorem, every n-vertex graph of maximum degree at most Delta >= 3 and clique number at most Delta is Delta-colorable, and thus it has an independent set of size at least n/Delta. We give an approximate characterization of graphs with independence number close to this bound, and use it to show that the problem of deciding whether such a graph has an independent set of size at least n/Delta+k has a kernel of size O(k).