Complexity of clique‐coloring odd‐hole‐free graphs

In this paper we investigate the problem of clique‐coloring, which consists in coloring the vertices of a graph in such a way that no monochromatic maximal clique appears, and we focus on odd‐hole‐free graphs. On the one hand we do not know any odd‐hole‐free graph that is not 3‐clique‐colorable, but on the other hand it is NP‐hard to decide if they are 2‐clique‐colorable, and we do not know if there exists any bound k₀ such that they are all k₀ ‐clique‐colorable. First we will prove that (odd hole, codiamond)‐free graphs are 2‐clique‐colorable. Then we will demonstrate that the complexity of 2‐clique‐coloring odd‐hole‐free graphs is actually Σ₂ P‐complete. Finally we will study the complexity of deciding whether or not a graph and all its subgraphs are 2‐clique‐colorable. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 139–156, 2009     

Clique‐coloring some classes of odd‐hole‐free graphs

We consider the problem of clique‐coloring, that is coloring the vertices of a given graph such that no maximal clique of size at least 2 is monocolored. Whereas we do not know any odd‐hole‐free graph that is not 3‐clique‐colorable, the existence of a constant C such that any perfect graph is C‐clique‐colorable is an open problem. In this paper we solve this problem for some subclasses of odd‐hole‐free graphs: those that are diamond‐free and those that are bull‐free. We also prove the NP‐completeness of 2‐clique‐coloring K₄‐free perfect graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 233–249, 2006     

Nowhere‐zero 3‐flows in locally connected graphs

Let G be a graph. For each vertex v ∈V(G), N_v denotes the subgraph induces by the vertices adjacent to v in G. The graph G is locally k‐edge‐connected if for each vertex v ∈V(G), N_v is k‐edge‐connected. In this paper we study the existence of nowhere‐zero 3‐flows in locally k‐edge‐connected graphs. In particular, we show that every 2‐edge‐connected, locally 3‐edge‐connected graph admits a nowhere‐zero 3‐flow. This result is best possible in the sense that there exists an infinite family of 2‐edge‐connected, locally 2‐edge‐connected graphs each of which does not have a 3‐NZF. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 211–219, 2003     

Sufficient conditions for graphs to be λ′‐optimal,super‐edge‐connected,and maximally edge‐connected

The restricted‐edge‐connectivity of a graph G, denoted by λ′(G), is defined as the minimum cardinality over all edge‐cuts S of G, where G‐S contains no isolated vertices. The graph G is called λ′‐optimal, if λ′(G) = ξ(G), where ξ(G) is the minimum edge‐degree in G. A graph is super‐edge‐connected, if every minimum edge‐cut consists of edges adjacent to a vertex of minimum degree. In this paper, we present sufficient conditions for arbitrary, triangle‐free, and bipartite graphs to be λ′‐optimal, as well as conditions depending on the clique number. These conditions imply super‐edge‐connectivity, if δ (G) ≥ 3, and the equality of edge‐connectivity and minimum degree. Different examples will show that these conditions are best possible and independent of other results in this area. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 228–246, 2005     

Minimally (k,k)‐edge‐connected graphs

For an integer l > 1, the l‐edge‐connectivity of a connected graph with at least l vertices is the smallest number of edges whose removal results in a graph with l components. A connected graph G is (k, l)‐edge‐connected if the l‐edge‐connectivity of G is at least k. In this paper, we present a structural characterization of minimally (k, k)‐edge‐connected graphs. As a result, former characterizations of minimally (2, 2)‐edge‐connected graphs in [J of Graph Theory 3 (1979), 15–22] are extended. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 116–131, 2003     

3‐Flows and Combs

Tutte's 3‐Flow Conjecture states that every 2‐edge‐connected graph with no 3‐cuts admits a 3‐flow. The 3‐Flow Conjecture is equivalent to the following: let G be a 2‐edge‐connected graph, let S be a set of at most three vertices of G; if every 3‐cut of G separates S then G has a 3‐flow. We show that minimum counterexamples to the latter statement are 3‐connected, cyclically 4‐connected, and cyclically 7‐edge‐connected.     

Spaces of Bessel‐potential type and embeddings: the super‐limiting case

We consider Bessel‐potential spaces modelled upon Lorentz‐Karamata spaces and establish embedding theorems in the super‐limiting case. In addition, we refine a result due to Triebel, in the context of Bessel‐potential spaces, itself an improvement of the Brézis‐Wainger result (super‐limiting case) about the “almost Lipschitz continuity” of elements of H^1+n/p_p (?ⁿ). These results improve and extend results due to Edmunds, Gurka and Opic in the context of logarithmic Bessel potential spaces. We also give examples of embeddings of Besselpotential type spaces which are not of logarithmic type. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical and asymptotic study of non‐axisymmetric magnetohydrodynamic boundary layer stagnation‐point flows

Both numerical and asymptotic analyses are performed to study the similarity solutions of three‐dimensional boundary‐layer viscous stagnation point flow in the presence of a uniform magnetic field. The three‐dimensional boundary‐layer is analyzed in a non‐axisymmetric stagnation point flow, in which the flow is developed because of influence of both applied magnetic field and external mainstream flow. Two approaches for the governing equations are employed: the Keller‐box numerical simulations solving full nonlinear coupled system and a corresponding linearized system that is obtained under a far‐field behavior and in the limit of large shear‐to‐strain‐rate parameter (λ). From these two approaches, the flow phenomena reveals a rich structure of new family of solutions for various values of the magnetic number and λ. The various results for the wall stresses and the displacement thicknesses are presented along with some velocity profiles in both directions. The analysis discovered that the flow separation occurs in the secondary flow direction in the absence of magnetic field, and the flow separation disappears when the applied magnetic field is increased. The flow field is divided into a near‐field (due to viscous forces) and far‐field (due to mainstream flows), and the velocity profiles form because of an interaction between two regions. The magnetic field plays an important role in reducing the thickness of the boundary‐layer. A physical explanation for all observed phenomena is discussed. Copyright © 2017 John Wiley & Sons, Ltd.     

Evaluating flood extent mapping of two hydraulic models, 1D HEC‐RAS and 2D LISFLOOD‐FP in comparison with aerial imagery observations in Gorgan flood plain,Iran

We compared flood mapping techniques using a one‐dimensional (1D) hydraulic model HEC‐RAS and two‐dimensional (2D) LISFLOOD‐FP for a 10‐km reach of Gorgan River in Iran. Both models were run using the same hydrologic input data. The input into the models was a steady discharge of 90 cm, corresponds to a flood peak occurred on March 25, 2012. Flood maps generated using these two models were compared with an observed flood inundation map, using F‐statistic. The roughness coefficients of the models were calibrated by maximizing the value of the F‐statistic. Based on the F‐statistic, LISFLOOD‐FP gives a slightly better result (F = 0.69) than HEC‐RAS (F = 0.67). Visual comparison of the flood extents generated by the two models showed reasonably good agreement. Validation was done using a flood event occurred on May 31, 2014. The LISFLOOD‐FP model gave a better result for validation as well. The 2D model showed more consistency in comparison with the 1D model.     

T‐joins intersecting small edge‐cuts in graphs

In an earlier paper 3 , we studied cycles in graphs that intersect all edge‐cuts of prescribed sizes. Passing to a more general setting, we examine the existence of T‐joins in grafts that intersect all edge‐cuts whose size is in a given set A ?{1,2,3}. In particular, we characterize all the contraction‐minimal grafts admitting no T‐joins that intersect all edge‐cuts of size 1 and 2. We also show that every 3‐edge‐connected graft admits a T‐join intersecting all 3‐edge‐cuts. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 64–71, 2007     

Hamilton cycle and Hamilton path extendability of Cayley graphs on abelian groups

In this paper the concepts of Hamilton cycle (HC) and Hamilton path (HP) extendability are introduced. A connected graph Γ is n‐HC‐extendable if it contains a path of length n and if every such path is contained in some Hamilton cycle of Γ. Similarly, Γ is weakly n‐HP‐extendable if it contains a path of length n and if every such path is contained in some Hamilton path of Γ. Moreover, Γ is strongly n‐HP‐extendable if it contains a path of length n and if for every such path P there is a Hamilton path of Γ starting with P. These concepts are then studied for the class of connected Cayley graphs on abelian groups. It is proved that every connected Cayley graph on an abelian group of order at least three is 2‐HC‐extendable and a complete classification of 3‐HC‐extendable connected Cayley graphs of abelian groups is obtained. Moreover, it is proved that every connected Cayley graph on an abelian group of order at least five is weakly 4‐HP‐extendable. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory     

Some limit analysis of a three dimensional viscous compressible capillary model for plasma

In this study, we discuss some limit analysis of a viscous capillary model of plasma, which is expressed as a so‐called the compressible Navier‐Stokes‐Poisson‐Korteweg equation. First, the existence of global smooth solutions for the initial value problem to the compressible Navier‐Stokes‐Poisson‐Korteweg equation with a given Debye length λ and a given capillary coefficient κ is obtained. We also show the uniform estimates of global smooth solutions with respect to the Debye length λ and the capillary coefficient κ. Then, from Aubin lemma, we show that the unique smooth solution of the 3‐dimensional Navier‐Stokes‐Poisson‐Korteweg equations converges globally in time to the strong solution of the corresponding limit equations, as λ tends to zero, κ tends to zero, and λ and κ simultaneously tend to zero. Moreover, we also give the convergence rates of these limits for any given positive time one by one.     

A Characterization of b‐Perfect Graphs

A b‐coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b‐chromatic number of a graph G is the largest integer k such that G admits a b‐coloring with k colors. A graph is b‐perfect if the b‐chromatic number is equal to the chromatic number for every induced subgraph of G. We prove that a graph is b‐perfect if and only if it does not contain as an induced subgraph a member of a certain list of 22 graphs. This entails the existence of a polynomial‐time recognition algorithm and of a polynomial‐time algorithm for coloring exactly the vertices of every b‐perfect graph. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:95–122, 2012     

s‐Regular cubic graphs as coverings of the complete bipartite graph K3,3

A graph is s‐regular if its automorphism group acts freely and transitively on the set of s‐arcs. An infinite family of cubic 1‐regular graphs was constructed in [10], as cyclic coverings of the three‐dimensional Hypercube. In this paper, we classify the s‐regular cyclic coverings of the complete bipartite graph K_3,3 for each ≥ 1 whose fibre‐preserving automorphism subgroups act arc‐transitively. As a result, a new infinite family of cubic 1‐regular graphs is constructed. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 101–112, 2004     

Cyclic 4‐Colorings of Graphs on Surfaces

To attack the Four Color Problem, in 1880, Tait gave a necessary and sufficient condition for plane triangulations to have a proper 4‐vertex‐coloring: a plane triangulation G has a proper 4‐vertex‐coloring if and only if the dual of G has a proper 3‐edge‐coloring. A cyclic coloring of a map G on a surface F² is a vertex‐coloring of G such that any two vertices x and y receive different colors if x and y are incident with a common face of G. In this article, we extend the result by Tait to two directions, that is, considering maps on a nonspherical surface and cyclic 4‐colorings.     

Gliding motion of bacteria on power‐law slime

The present investigation deals with an undulating surface model for the motility of bacteria gliding on a layer of non‐Newtonian slime. The slime being the viscoelastic material is considered as a power‐law fluid. A hydrodynamical model of motility involving an undulating cell surface which transmits stresses through a layer of exuded slime to the substratum is examined. The non‐linear differential equation resulting from the balance of momentum and mass is solved numerically by a finite difference method with an iteration technique. The manner in which the various exponent values of the power‐law flow affect the structure of the boundary layer is delineated. A comparison is made of the power‐law fluid with the Newtonian fluid. For the power‐law fluid with respect to different power‐law exponent values, shear‐thinning and shear‐thickening effects can be observed, respectively. Copyright © 2004 John Wiley & Sons, Ltd.     

Even‐hole‐free graphs part I: Decomposition theorem

We prove a decomposition theorem for even‐hole‐free graphs. The decompositions used are 2‐joins and star, double‐star and triple‐star cutsets. This theorem is used in the second part of this paper to obtain a polytime recognition algorithm for even‐hole‐free graphs. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 6–49, 2002     

On the Cauchy problem for the two‐component b‐family system

In this paper, we establish the local well‐posedness for the two‐component b‐family system in a range of the Besov space. We also derive the blow‐up scenario for strong solutions of the system. In addition, we determine the wave‐breaking mechanism to the two‐component Dullin–Gottwald–Holm system. Copyright © 2013 John Wiley & Sons, Ltd.     

3‐colorability of 4‐regular hamiltonian graphs

On the model of the cycle‐plus‐triangles theorem, we consider the problem of 3‐colorability of those 4‐regular hamiltonian graphs for which the components of the edge‐complement of a given hamiltonian cycle are non‐selfcrossing cycles of constant length ≥ 4. We show that this problem is NP‐complete. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 125–140, 2003     

Excluding a bipartite circle graph from line graphs

We prove that, for a fixed bipartite circle graph H, all line graphs with sufficiently large rank‐width (or clique‐width) must have a pivot‐minor isomorphic to H. To prove this, we introduce graphic delta‐matroids. Graphic delta‐matroids are minors of delta‐matroids of line graphs and they generalize graphic and cographic matroids. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 183–203, 2009