Approximately coloring graphs without long induced paths

It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on $t$ vertices, for fixed $t$. We propose an algorithm that, given a 3-colorable graph without an induced path on $t$ vertices, computes a coloring with $\max\{5,2\lceil{\frac{t-1}{2}}\rceil-2\}$ many colors. If the input graph is triangle-free, we only need $\max\{4,\lceil{\frac{t-1}{2}}\rceil+1\}$ many colors. The running time of our algorithm is $O((3^{t-2}+t^2)m+n)$ if the input graph has $n$ vertices and $m$ edges

Magnetic domain-wall motion by propagating spin waves

We found by micromagnetic simulations that the motion of a transverse wall (TW) type domain wall in magnetic thin-film nanostripes can be manipulated via interaction with spin waves (SWs) propagating through the TW. The velocity of the TW motion can be controlled by changes of the frequency and amplitude of the propagating SWs. Moreover, the TW motion is efficiently driven by specific SW frequencies that coincide with the resonant frequencies of the local modes existing inside the TW structure. The use of propagating SWs, whose frequencies are tuned to those of the intrinsic TW modes, is an alternative approach for controlling TW motion in nanostripes

Pennsylvanian-Early Triassic stratigraphy in the Alborz Mountains (Iran)

New fieldwork was carried out in the central and eastern Alborz, addressing the sedimentary succession from the Pennsylvanian to the Early Triassic. A regional synthesis is proposed, based on sedimentary analysis and a wide collection of new palaeontological data. The Moscovian Qezelqaleh Formation, deposited in a mixed coastal marine and alluvial setting, is present in a restricted area of the eastern Alborz, transgressing on the Lower Carboniferous Mobarak and Dozdehband formations. The late Gzhelian–early Sakmarian Dorud Group is instead distributed over most of the studied area, being absent only in a narrow belt to the SE. The Dorud Group is typically tripartite, with a terrigenous unit in the lower part (Toyeh Formation), a carbonate intermediate part (Emarat and Ghosnavi formations, the former particularly rich in fusulinids), and a terrigenous upper unit (Shah Zeid Formation), which however seems to be confined to the central Alborz. A major gap in sedimentation occurred before the deposition of the overlying Ruteh Limestone, a thick package of packstone–wackestone interpreted as a carbonate ramp of Middle Permian age (Wordian–Capitanian). The Ruteh Limestone is absent in the eastern part of the range, and everywhere ends with an emersion surface, that may be karstified or covered by a lateritic soil. The Late Permian transgression was directed southwards in the central Alborz, where marine facies (Nesen Formation) are more common. Time-equivalent alluvial fans with marsh intercalations and lateritic soils (Qeshlaq Formation) are present in the east. Towards the end of the Permian most of the Alborz emerged, the marine facies being restricted to a small area on the Caspian side of the central Alborz. There, the Permo-Triassic boundary interval is somewhat similar to the Abadeh–Shahreza belt in central Iran, and contains oolites, flat microbialites and domal stromatolites, forming the base of the Elikah Formation. The P–T boundary is established on the basis of conodonts, small foraminifera and stable isotope data. The development of the lower and middle part of the Elikah Formation, still Early Triassic in age, contains vermicular bioturbated mudstone/wackestone, and anachronostic-facies-like gastropod oolites and flat pebble conglomerates. Three major factors control the sedimentary evolution. The succession is in phase with global sea-level curve in the Moscovian and from the Middle Permian upwards. It is out of phase around the Carboniferous–Permian boundary, when the Dorud Group was deposited during a global lowstand of sealevel. When the global deglaciation started in the Sakmarian, sedimentation stopped in the Alborz and the area emerged. Therefore, there is a consistent geodynamic control. From the Middle Permian upwards, passive margin conditions control the sedimentary evolution of the basin, which had its depocentre(s) to the north. Climate also had a significant role, as the Alborz drifted quickly northwards with other central Iran blocks towards the Turan active margin. It passed from a southern latitude through the aridity belt in the Middle Permian, across the equatorial humid belt in the Late Permian and reached the northern arid tropical belt in the Triassic

Complexity of Coloring Graphs without Paths and Cycles

Let $P_t$ and $C_\ell$ denote a path on $t$ vertices and a cycle on $\ell$ vertices, respectively. In this paper we study the $k$-coloring problem for $(P_t,C_\ell)$-free graphs. Maffray and Morel, and Bruce, Hoang and Sawada, have proved that 3-colorability of $P_5$-free graphs has a finite forbidden induced subgraphs characterization, while Hoang, Moore, Recoskie, Sawada, and Vatshelle have shown that $k$-colorability of $P_5$-free graphs for $k \geq 4$ does not. These authors have also shown, aided by a computer search, that 4-colorability of $(P_5,C_5)$-free graphs does have a finite forbidden induced subgraph characterization. We prove that for any $k$, the $k$-colorability of $(P_6,C_4)$-free graphs has a finite forbidden induced subgraph characterization. We provide the full lists of forbidden induced subgraphs for $k=3$ and $k=4$. As an application, we obtain certifying polynomial time algorithms for 3-coloring and 4-coloring $(P_6,C_4)$-free graphs. (Polynomial time algorithms have been previously obtained by Golovach, Paulusma, and Song, but those algorithms are not certifying); To complement these results we show that in most other cases the $k$-coloring problem for $(P_t,C_\ell)$-free graphs is NP-complete. Specifically, for $\ell=5$ we show that $k$-coloring is NP-complete for $(P_t,C_5)$-free graphs when $k \ge 4$ and $t \ge 7$; for $\ell \ge 6$ we show that $k$-coloring is NP-complete for $(P_t,C_\ell)$-free graphs when $k \ge 5$, $t \ge 6$; and additionally, for $\ell=7$, we show that $k$-coloring is also NP-complete for $(P_t,C_7)$-free graphs if $k = 4$ and $t\ge 9$. This is the first systematic study of the complexity of the $k$-coloring problem for $(P_t,C_\ell)$-free graphs. We almost completely classify the complexity for the cases when $k \geq 4, \ell \geq 4$, and identify the last three open cases

Exhaustive generation of kk-critical H\mathcal H-free graphs

We describe an algorithm for generating all $k$-critical $\mathcal H$-free graphs, based on a method of Ho\`{a}ng et al. Using this algorithm, we prove that there are only finitely many $4$-critical $(P_7,C_k)$-free graphs, for both $k=4$ and $k=5$. We also show that there are only finitely many $4$-critical graphs $(P_8,C_4)$-free graphs. For each case of these cases we also give the complete lists of critical graphs and vertex-critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the $3$-colorability problem in the respective classes. Moreover, we prove that for every $t$, the class of 4-critical planar $P_t$-free graphs is finite. We also determine all 27 4-critical planar $(P_7,C_6)$-free graphs. We also prove that every $P_{10}$-free graph of girth at least five is 3-colorable, and determine the smallest 4-chromatic $P_{12}$-free graph of girth five. Moreover, we show that every $P_{13}$-free graph of girth at least six and every $P_{16}$-free graph of girth at least seven is 3-colorable. This strengthens results of Golovach et al.Comment: 17 pages, improved girth results. arXiv admin note: text overlap with arXiv:1504.0697

Direct observation of domain wall structures in curved permalloy wires containing an antinotch

The formation and field response of head-to-head domain walls in curved permalloy wires, fabricated to contain a single antinotch, have been investigated using Lorentz microscopy. High spatial resolution maps of the vector induction distribution in domain walls close to the antinotch have been derived and compared with micromagnetic simulations. In wires of 10 nm thickness the walls are typically of a modified asymmetric transverse wall type. Their response to applied fields tangential to the wire at the antinotch location was studied. The way the wall structure changes depends on whether the field moves the wall away from or further into the notch. Higher fields are needed and much more distorted wall structures are observed in the latter case, indicating that the antinotch acts as an energy barrier for the domain wal

List coloring in the absence of a linear forest.

The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The Listk-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u)⊆{1,…,k}. Let Pn denote the path on n vertices, and G+H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that Listk-Coloring can be solved in polynomial time for graphs with no induced rP1+P5, hereby extending the result of Hoàng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P5. Our result is tight; we prove that for any graph H that is a supergraph of P1+P5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H

On the computation of zone and double zone diagrams

Classical objects in computational geometry are defined by explicit relations. Several years ago the pioneering works of T. Asano, J. Matousek and T. Tokuyama introduced "implicit computational geometry", in which the geometric objects are defined by implicit relations involving sets. An important member in this family is called "a zone diagram". The implicit nature of zone diagrams implies, as already observed in the original works, that their computation is a challenging task. In a continuous setting this task has been addressed (briefly) only by these authors in the Euclidean plane with point sites. We discuss the possibility to compute zone diagrams in a wide class of spaces and also shed new light on their computation in the original setting. The class of spaces, which is introduced here, includes, in particular, Euclidean spheres and finite dimensional strictly convex normed spaces. Sites of a general form are allowed and it is shown that a generalization of the iterative method suggested by Asano, Matousek and Tokuyama converges to a double zone diagram, another implicit geometric object whose existence is known in general. Occasionally a zone diagram can be obtained from this procedure. The actual (approximate) computation of the iterations is based on a simple algorithm which enables the approximate computation of Voronoi diagrams in a general setting. Our analysis also yields a few byproducts of independent interest, such as certain topological properties of Voronoi cells (e.g., that in the considered setting their boundaries cannot be "fat").Comment: Very slight improvements (mainly correction of a few typos); add DOI; Ref [51] points to a freely available computer application which implements the algorithms; to appear in Discrete & Computational Geometry (available online

Domain wall propagation in Permalloy nanowires with a thickness gradient

The domain wall nucleation and motion processes in Permalloy nanowires with a thickness gradient along the nanowire axis have been studied. Nanowires with widths, w = 250 nm to 3 um and a base thickness of t = 10 nm were fabricated by electron-beam lithography. The magnetization hysteresis loops measured on individual nanowires are compared to corresponding nanowires without a thickness gradient. The Hc vs. t/w curves of wires with and without a thickness gradient are discussed and compared to micromagnetic simulations. We find a metastability regime at values of w, where a transformation from transverse to vortex domain wall type is expected

Parameterized Complexity of Maximum Edge Colorable Subgraph

A graph $H$ is {\em $p$-edge colorable} if there is a coloring $\psi: E(H) \rightarrow \{1,2,\dots,p\}$, such that for distinct $uv, vw \in E(H)$, we have $\psi(uv) \neq \psi(vw)$. The {\sc Maximum Edge-Colorable Subgraph} problem takes as input a graph $G$ and integers $l$ and $p$, and the objective is to find a subgraph $H$ of $G$ and a $p$-edge-coloring of $H$, such that $|E(H)| \geq l$. We study the above problem from the viewpoint of Parameterized Complexity. We obtain \FPT\ algorithms when parameterized by: $(1)$ the vertex cover number of $G$, by using {\sc Integer Linear Programming}, and $(2)$ $l$, a randomized algorithm via a reduction to \textsc{Rainbow Matching}, and a deterministic algorithm by using color coding, and divide and color. With respect to the parameters $p+k$, where $k$ is one of the following: $(1)$ the solution size, $l$, $(2)$ the vertex cover number of $G$, and $(3)$ l - {\mm}(G), where {\mm}(G) is the size of a maximum matching in $G$; we show that the (decision version of the) problem admits a kernel with $\mathcal{O}(k \cdot p)$ vertices. Furthermore, we show that there is no kernel of size $\mathcal{O}(k^{1-\epsilon} \cdot f(p))$, for any $\epsilon > 0$ and computable function $f$, unless \NP \subseteq \CONPpoly