Planar Hall effect in biosensor with a tilted angle of the cross-junction

A typical planar Hall effect (PHE) sensor junction consists of two Hall bars that the bars appear normal to each other and the junction can have the required four terminals for current and voltage measurements. We are now introducing a tilted angle of the cross-junction and studying the role of the PHE therein. The results show that although there is a tilted angle of the cross-junction, the PHE voltage is remained constant. The result is interpreted by assuming the sensor material with the behavior of a basic single domain structure under the external magnetic field reversals. The calculations of the model are found to be in good concurrence with the experimental results.     

An imaging pH optode for cell studies based on covalent attachment of 8-hydroxypyrene-1,3,6-trisulfonate to amino cellulose acetate films

The objective of this study was to create a thin film optode for fast pH measurements that meets the requirements for imaging pH-responses from cells as well as for a future hybrid design for detection of multiple analytes simultaneously. The sensor is based on the covalent attachment of 8-hydroxypyrene-1,3,6-trisulfonate (HPTS) to a film forming cellulose acetate material through a sulfonamide linkage. The synthesis routes of the cellulose material and regio-specific covalent attachment of the dye are described in detail. The sensor was sterilized in two different ways and showed excellent biocompatibility with Chinese hamster ovary cells. Imprints from cells and μm thin cell extensions were visualized when altering pH of the surrounding solution. The sensor was tested together with time-dependent sigmoidal calibrations giving pH determinations with an exceptional precision and accuracy during measurement within pH 6-8. The precision of the optode, calculated as pooled S.D. (n = 8) according to IUPAC recommendations between pH 6.641 and 7.742 was 0.0029. The accuracy was significantly better than the electrode used as reference during the measurements. The response time (0-95%) was 100 s between pH 6.641 and 7.742 and the reverse response (95-0%) was 80 s. The sensor shows great potential for extra-cellular pH determination over time during cell growth and pharmacological exposure.     

Edge-choosability of planar graphs without adjacent triangles or without 7-cycles

A graph G is edge-L-colorable, if for a given edge assignment L={L(e):e∈E(G)}, there exists a proper edge-coloring ? of G such that ?(e)∈L(e) for all e∈E(G). If G is edge-L-colorable for every edge assignment L with |L(e)|≥k for e∈E(G), then G is said to be edge-k-choosable. In this paper, we prove that if G is a planar graph with maximum degree Δ(G)≠5 and without adjacent 3-cycles, or with maximum degree Δ(G)≠5,6 and without 7-cycles, then G is edge-(Δ(G)+1)-choosable.     

Equitable list coloring of planar graphs without 4- and 6-cycles

A graph G is equitably k-choosable if for any k-uniform list assignment L, there exists an L-colorable of G such that each color appears on at most vertices. Kostochka, Pelsmajer and West introduced this notion and conjectured that G is equitably k-choosable for k>Δ(G). We prove this for planar graphs with Δ(G)≥6 and no 4- or 6-cycles.     

Polychromatic Colorings of Plane Graphs

We show that the vertices of any plane graph in which every face is incident to at least g vertices can be colored by ⌊(3g−5)/4⌋ colors so that every color appears in every face. This is nearly tight, as there are plane graphs where all faces are incident to at least g vertices and that admit no vertex coloring of this type with more than ⌊(3g+1)/4⌋ colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by k colors in which all colors appear in every face is in ℘ for k=2 and is -complete for k=3,4. We refine this result for polychromatic 3-colorings restricted to 2-connected graphs which have face sizes from a prescribed (possibly infinite) set of integers. Thereby we find an almost complete characterization of these sets of integers (face sizes) for which the corresponding decision problem is in ℘, and for the others it is -complete. Research of N. Alon was supported in part by the Israel Science Foundation, by a USA–Israeli BSF grant, and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. Research of R. Berke was supported in part by JSPS Global COE program “Computationism as a Foundation for the Sciences.” Research of K. Buchin and M. Buchin was supported by the Netherlands’ Organisation for Scientific Research (NWO) under BRICKS/FOCUS project no. 642.065.503. Research of P. Csorba was supported by DIAMANT, an NWO mathematics cluster. Research of B. Speckmann was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.022.707.     

Weighted coloring on planar, bipartite and split graphs: Complexity and approximation

We study complexity and approximation of min weighted node coloring in planar, bipartite and split graphs. We show that this problem is NP-hard in planar graphs, even if they are triangle-free and their maximum degree is bounded above by 4. Then, we prove that min weighted node coloring is NP-hard in P₈-free bipartite graphs, but polynomial for P₅-free bipartite graphs. We next focus on approximability in general bipartite graphs and improve earlier approximation results by giving approximation ratios matching inapproximability bounds. We next deal with min weighted edge coloring in bipartite graphs. We show that this problem remains strongly NP-hard, even in the case where the input graph is both cubic and planar. Furthermore, we provide an inapproximability bound of 7/6−ε, for any ε>0 and we give an approximation algorithm with the same ratio. Finally, we show that min weighted node coloring in split graphs can be solved by a polynomial time approximation scheme.     

Planar graphs with maximum degree 8 and without adjacent triangles are 9-totally-colorable

The Total Coloring Conjecture, in short, TCC, says that every simple graph is (Δ+2)-totally-colorable where Δ is the maximum degree of the graph. Even for planar graphs this conjecture has not been completely settled yet. However, every planar graph with Δ≥9 has been proved to be (Δ+1)-totally-colorable. In this paper, we prove that planar graphs with maximum degree 8 and without adjacent triangles are 9-totally-colorable.     

The vertex-transitive TLF-planar graphs

We consider the class of the topologically locally finite (in short TLF) planar vertex-transitive graphs. We characterize these graphs by finite combinatorial objects called labeling schemes. As a result, we are able to enumerate and describe all TLF-planar vertex-transitive graphs of given degree, as well as most of their transitive groups of automorphisms. In addition,we are able to decide whether a given TLF-planar transitive graph is Cayley or not. This class contains all the one-ended planar Cayley graphs and the normal transitive tilings of the plane.     

Planar graphs without 5-cycles or without 6-cycles

Let G be a planar graph without 5-cycles or without 6-cycles. In this paper, we prove that if G is connected and δ(G)≥2, then there exists an edge xy∈E(G) such that d(x)+d(y)≤9, or there is a 2-alternating cycle. By using the above result, we obtain that (1) its linear 2-arboricity , (2) its list total chromatic number is Δ(G)+1 if Δ(G)≥8, and (3) its list edge chromatic number is Δ(G) if Δ(G)≥8.     

An oriented coloring of planar graphs with girth at least five

An oriented k-coloring of an oriented graph G is a homomorphism from G to an oriented graph H of order k. We prove that every oriented graph with a maximum average degree less than and girth at least 5 has an oriented chromatic number at most 16. This implies that every oriented planar graph with girth at least 5 has an oriented chromatic number at most 16, that improves the previous known bound of 19 due to Borodin et al. [O.V. Borodin, A.V. Kostochka, J. Nešet?il, A. Raspaud, É. Sopena, On the maximum average degree and the oriented chromatic number of a graph, Discrete Math. 206 (1999) 77-89].