Colorable Graphs Research Articles

On topological classification of Morse–Smale diffeomorphisms on the sphere S n (n > 3)

We consider the class G(S n ) of orientation preserving Morse–Smale diffeomorphisms of the sphere S n of dimension n > 3, assuming that invariant manifolds of different saddle periodic points have no intersection. For any diffeomorphism f ∈ G(S n ), we define a coloured graph Γ f that describes a mutual arrangement of invariant manifolds of saddle periodic points of the diffeomorphism f. We enrich the graph Γ f by an automorphism P f induced by dynamics of f and define the isomorphism notion between two coloured graphs. The aim of the paper is to show that two diffeomorphisms f, f′ ∈ G(S n ) are topologically conjugated if and only if the graphs Γ f , Γ f ′ are isomorphic. Moreover, we establish the existence of a linear-time algorithm to distinguish coloured graphs of diffeomorphisms from the class G(S n ).

Comprehensive analysis of important parameters of choline chloride-based deep eutectic solvent pretreatment of lignocellulosic biomass

Choline chloride based deep eutectic solvents have showed great potential in lignocellulosic biomass pretreatment. In this study, for DES pretreatment with different hydrogen bond donners of different raw materials under different reaction conditions, multivariate analysis methods including principal component analysis and partial least squares analysis were used for reveal the pretreatment mechanism by evaluating the inner relationships among 42 key process factors. Furthermore, based on molecular simulation, the detailed relationships between key variables were further analyzed. Meanwhile, four-dimensional color graphs were used to intuitively reveal the synergistic influence of multivariate conditions variables on pretreatment effect to obtain better economic benefits and energy consumption indicators for DES pretreatment. The results showed that HBD hydrophilic ability, HBD polarity, HBD acidity, HBD ability to form hydrogen bonds, molar ratio of HBD to choline chloride and pretreatment severity had great influence on the Choline chloride based deep eutectic solvents pretreatment effect.

Micropolar pulsatile blood flow conveying nanoparticles in a stenotic tapered artery: NON-Newtonian pharmacodynamic simulation

Two-dimensional rheological laminar hemodynamics through a diseased tapered artery with a mild stenosis present is simulated theoretically and computationally. The effect of different metallic nanoparticles homogeneously suspended in the blood is considered, motivated by drug delivery (pharmacology) applications. The Eringen micropolar model has been discussed for hemorheological characteristics in the whole arterial region. The conservation equations for mass, linear momentum, angular momentum (micro-rotation), and energy and nanoparticle species are normalized by employing suitable non-dimensional variables. The transformed equations are solved numerically subject to physically appropriate boundary conditions using the finite element method with the variational formulation scheme available in the FreeFEM++ code. A good correlation is achieved between the FreeFEM++ computations and existing results. The effect of selected parameters (taper angle, Prandtl number, Womersley parameter, pulsatile constants, and volumetric concentration) on velocity, temperature, and micro-rotational (Eringen angular) velocity has been calculated for a stenosed arterial segment. Wall shear stress, volumetric flow rate, and hemodynamic impedance of blood flow are also computed. Colour contours and graphs are employed to visualize the simulated blood flow characteristics. It is observed that by increasing Prandtl number (Pr), the micro-rotational velocity decreases i.e., microelement (blood cell) spin is suppressed. Wall shear stress decreases with the increment in pulsatile parameters (B and e), whereas linear velocity increases with a decrement in these parameters. Furthermore, the velocity decreases in the tapered region with elevation in the Womersley parameter (α). The simulations are relevant to transport phenomena in pharmacology and nano-drug targeted delivery in hematology.

Coloured random graphs explain the structure and dynamics of cross-linked polymer networks

Step-growth and chain-growth are two major families of chemical reactions that result in polymer networks with drastically different physical properties, often referred to as hyper-branched and cross-linked networks. In contrast to step-growth polymerisation, chain-growth forms networks that are history-dependent. Such networks are defined not just by the degree distribution, but also by their entire formation history, which entails a modelling and conceptual challenges. We show that the structure of chain-growth polymer networks corresponds to an edge-coloured random graph with a defined multivariate degree distribution, where the colour labels represent the formation times of chemical bonds. The theory quantifies and explains the gelation in free-radical polymerisation of cross-linked polymers and predicts conditions when history dependance has the most significant effect on the global properties of a polymer network. As such, the edge colouring is identified as the key driver behind the difference in the physical properties of step-growth and chain-growth networks. We expect that this findings will stimulate usage of network science tools for discovery and design of cross-linked polymers.

Combinatorial Nullstellensatz and DP-coloring of graphs

We initiate the study of applying the Combinatorial Nullstellensatz to the DP-coloring of graphs even though, as is well-known, the Alon–Tarsi theorem does not apply to DP-coloring. We define the notion of good covers of prime order which allows us to apply the Combinatorial Nullstellensatz to DP-coloring. We apply these tools to DP-coloring of the cones of certain bipartite graphs and uniquely 3-colorable graphs. We also extend a result of Akbari et al. (2006) on unique list colorability to the context of DP-coloring. We establish a sufficient algebraic condition for a graph G to satisfy χDP(G)≤3, and we completely determine the DP-chromatic number of squares of all cycles.

Rainbow Matchings of size $m$ in Graphs with Total Color Degree at least $2mn$

The existence of a rainbow matching given a minimum color degree, proper coloring, or triangle-free host graph has been studied extensively. This paper generalizes these problems to edge colored graphs with given total color degree. In particular, we find that if a graph $G$ has total color degree $2mn$ and satisfies some other properties, then $G$ contains a matching of size $m$. These other properties include $G$ being triangle-free, $C_4$-free, properly colored, or large enough.

Mechanical and Microstructural Characterization of Friction Stir-Welded AA2219: TiCp-Reinforced Composites

Light materials having high strength-to-weight ratio plays a vital role in material science especially in the field of aerospace, marine transportation, military and structural applications. Composite serves the same. Aluminium matrix composite (AMC) is the aluminium matrix which has high strength-to-weight ratio. The purpose of the work is to investigate the mechanical properties of welded AA2219 aluminium alloy reinforced with titanium carbide particles (TiCp). Aluminium matrix is reinforced with TiCp using modified stir casting process. Optical colour graphs, XRD and EDX analysis reveal the presence of TiCp in cast AMC. Tensile test shows the ultimate tensile strength increases with the percentage of TiCp addition. Cut samples of size 100 mm X 50 mm X 6 mm cast AMC were friction stir welded using response surface modelling. Ultimate tensile strength and joint efficiency were calculated for the weld samples, and the effects of input parameters were investigated. Experimental results indicate that tool pin profile has a significant effect on tensile strength of friction stir-welded joints followed by welding speed. It is evident from the response that tool rotational speed had the least effect on ultimate tensile strength. The calculated joint efficiency of the most weld samples was more than 90% with respect to the ultimate tensile strength of the base metal. This work ensures defect free, high efficiency welded joints produced using a wide range of process parameters and recommends parameters for producing best joint tensile properties.

Assessment of arable lands use in the VEGA satellite monitoring services on the example of Primorsky Krai, Russia

The paper considers the possibility of arable lands use assessing by the facilities of satellite monitoring services VEGA. The services provide access to archived and operational satellite data of various spectral bands, spatial resolution and time ranges. They also implement various tools for interactive and automated processing of this data. These include comparative visual analysis of multi-temporal satellite imagery, visual analysis of multi-temporal color composites and graphs of vegetation indices dynamics, pixel-based and object-based classification. The main features of cultivated and abandoned arable lands are considered on the example of Primorsky Krai, Russia. It is shown that the object-based classification tool implemented in the services allows user to classify cultivated and abandoned arable lands with an accuracy above 90%.

Colorings of the Graph K ᵐ 2 + Kn

In this paper, we characterize chromatically unique, determine list-chromatic number and characterize uniquely list colorability of the graph G = Km 2 + Kn. We shall prove that G is χ-unique, ch(G) = m + n, G is uniquely 3-list colorable graph if and only if 2m + n > 7 and m > 2

A Generalization of Some Results on List Coloring and DP-Coloring

Let G be a graph and let $$f_i, i \in \{1,\ldots ,s\},$$ be a function from V(G) to the set of nonnegative integers. In Sittitrai and Nakprasit (Analogue of DP-coloring on variable degeneracy and its applications, 2020), the concept of DP-F-coloring, a generalization of DP-coloring and variable degeneracy, was introduced. We use DP-F-coloring to define DPG-[k, t]-colorable graphs and modify the proofs in Liu et al. (Graphs Combin 35(3), 695–705, 2019), Sittitrai and Nakprasit (Bull Malays Math Sci Soc, 2019, https://doi.org/10.1007/s40840-019-00800-1), Thomassen (J Combin Theory Ser B 62, 180–181, 1994) to obtain more results on list coloring, DP-coloring, list-forested coloring, and variable degeneracy.

Large monochromatic components in 3-colored non-complete graphs

We show that in every 3-coloring of the edges of a graph G of order N such that δ(G)≥5N6−1, there is a monochromatic component of order at least N/2. We also show that this result is best possible.

On the total and AVD-total coloring of graphs

A total coloring of a graph G is an assignment of colors to the vertices and the edges such that (i) no two adjacent vertices receive same color, (ii) no two adjacent edges receive same color, and (iii) if an edge e is incident on a vertex v, then v and e receive different colors. The least number of colors sufficient for a total coloring of graph G is called its total chromatic number and denoted by An adjacent vertex distinguishing (AVD)-total coloring of G is a total coloring with the additional property that for any adjacent vertices u and v, the set of colors used on the edges incident on u including the color of u is different from the set of colors used on the edges incident on v including the color of v. The adjacent vertex distinguishing (AVD)-total chromatic number of G, is the minimum number of colors required for a valid AVD-total coloring of G. It is conjectured that which is known as total coloring conjecture and is one of the famous open problems. A graph for which the total coloring conjecture holds is called totally colorable graph. The problem of deciding whether or for a totally colorable graph G is called the classification problem for total coloring. However, this classification problem is known to be NP-hard even for bipartite graphs. In this paper, we give a sufficient condition for a bipartite biconvex graph G to have Also, we propose a linear time algorithm to compute the total chromatic number of chain graphs, a proper subclass of biconvex graphs. We prove that the total coloring conjecture holds for the central graph of any graph. Finally, we obtain the AVD-total chromatic number of central graphs for basic graphs such as paths, cycles, stars and complete graphs.

On Restricted Colorings of text{(d,s)}-Edge Colorable Graphs

A cycle is 2-colored if its edges are properly colored by two distinct colors. A (d, s)-edge colorable graphG is a d-regular graph that admits a proper d-edge coloring in which every edge of G is in at least s-1 2-colored 4-cycles. Given a (d, s)-edge colorable graph G and a list assigment L of forbidden colors for the edges of G satisfying certain sparsity conditions, we prove that there is a proper d-edge coloring of G that avoids L, that is, a proper edge coloring varphi of G such that varphi (e) notin L(e) for every edge e of G. Additionally, this paper also contains a discussion of graphs belonging to the family of (d, s)-edge colorable graphs.

Computational fluid dynamic simulation of two-fluid non-Newtonian nanohemodynamics through a diseased artery with a stenosis and aneurysm

This article presents a two-dimensional theoretical study of hemodynamics through a diseased permeable artery with a mild stenosis and an aneurysm present. The effect of metallic nanoparticles on the blood flow is considered, motivated by drug delivery (pharmacology) applications. Two different models are adopted to mimic non-Newtonian characteristics of the blood flow; the Casson (viscoplastic) fluid model is deployed in the core region and the Sisko (viscoelastic) fluid model employed in the peripheral (porous) region. The revised Buongiorno two-component nanofluid model is utilized for nanoscale effects. The blood is considered to contain a homogenous suspension of nanoparticles. The governing equations are derived by extending the Navier-Stokes equations with linear Boussinesq approximation (which simulates both heat and mass transfer). Natural (free) double-diffusive convection is considered to simulate the dual influence of thermal and solutal buoyancy forces. The conservation equations are normalised by employing appropriate non-dimensional variables. The transformed equations are solved numerically using the finite element method with the variational formulation scheme available in the FreeFEM++ code. A comprehensive mesh-independence study is included. The effect of selected parameters (thermophoresis, Brownian motion, Grashof number, thermo-solutal buoyancy ratio, Sisko parameter ratio, and permeability parameter) on velocity, temperature, nanoparticle concentration, and hemodynamic pressure have been calculated for two clinically important cases of arteries with stenosis and an aneurysm. Skin-friction coefficient, Nusselt number, volumetric flow rate, and resistance impedance of blood flow are also computed. Colour contours and graphs are employed to visualize the simulated blood flow characteristics. It is observed that by increasing the thermal buoyancy parameter, i.e. Grashof number (Gr), the nanoparticle concentration and temperature decrease, whereas velocity increases with an increment in the Brownian motion parameter (Nb). Furthermore, velocity decreases in the peripheral porous region with elevation in the Sisko material ratio (m) and permeability parameter (k’). The simulations are relevant to transport phenomena in pharmacology and nano-drug targeted delivery in haematology.

Graphs in which all maximal bipartite subgraphs have the same order

Motivated by the concept of well-covered graphs, we define a graph to be well-bicovered if every vertex-maximal bipartite subgraph has the same order (which we call the bipartite number). We first give examples of them, compare them with well-covered graphs, and characterize those with small or large bipartite number. We then consider graph operations including the union, join, and lexicographic and cartesian products. Thereafter we consider simplicial vertices and 3-colored graphs where every vertex is in triangle, and conclude by characterizing the maximal outerplanar graphs that are well-bicovered.

On garlands in $\chi$-uniquely colorable graphs

On garlands in $\chi$-uniquely colorable graphs

On Uniquely 3-Colorable Plane Graphs without Adjacent Faces of Prescribed Degrees

A graph G is uniquely k-colorable if the chromatic number of G is k and G has only one k-coloring up to the permutation of the colors. For a plane graph G, two faces f 1 and f 2 of G are adjacent ( i , j )-faces if d ( f 1 ) = i, d ( f 2 ) = j, and f 1 and f 2 have a common edge, where d ( f ) is the degree of a face f. In this paper, we prove that every uniquely three-colorable plane graph has adjacent ( 3 , k )-faces, where k ≤ 5. The bound of five for k is the best possible. Furthermore, we prove that there exists a class of uniquely three-colorable plane graphs having neither adjacent ( 3 , i )-faces nor adjacent ( 3 , j )-faces, where i , j are fixed in { 3 , 4 , 5 } and i ≠ j. One of our constructions implies that there exists an infinite family of edge-critical uniquely three-colorable plane graphs with n vertices and 7 3 n - 14 3 edges, where n ( ≥ 11 ) is odd and n ≡ 2 ( mod 3 ).

Some properties of Bowlin and Brin’s color graphs

Bowlin and Brin defined the class of color graphs, whose vertices are triangulated polygons compatible with a fixed four-coloring of the polygon vertices. In this article it is proven that each color graph has a vertex-induced embedding in a hypercube, and an upper bound is given for the hypercube dimension. The color graphs for n-gons up to n=8 are listed and studied, in particular enabling a question by Bowlin and Brin concerning the diameter of color graphs to be answered. Finally it is shown that color graphs with a certain type of subgraph cannot be isometrically embedded in a hypercube of any dimension.

Chromatic Number and Orientations of Graphs and Signed Graphs

Assume $D$ is a digraph, and $D'$ is a spanning sub-digraph of $D$. We say $D'$ is a modulo-$k$ Eulerian sub-digraph of $D$ if for each vertex $v$ of $D'$, $d_{D'}^+(v) \equiv d_{D'}^-(v) \pmod{k}$. A modulo-$k$ Eulerian sub-digraph $D'$ of $D$ is special if for every vertex $v$, $d_D^+(v) = 0$ implies $d_{D'}^-(v) = 0$ and $d_{D'}^+(v) = d_D^+(v) > 0$ implies $d_{D'}^-(v) > 0$. We denote by $\operatorname{OE}_k(D)$ or $\operatorname{EE}_k(D)$ (respectively, $\operatorname{OE}_k^s(D)$ or $\operatorname{EE}_k^s(D)$) the sets of spanning modulo-$k$ Eulerian sub-digraphs (respectively, the sets of spanning special modulo-$k$ Eulerian sub-digraphs) of $D$ with an odd number or even number of edges. Matiyasevich [A criterion for vertex colorability of a graph stated in terms of edge orientations, (in Russia), Diskretnyi Analiz, issue 26, 65--71 (1974)] proved that a graph $G$ is $k$-colourable if and only if $G$ has an orientation $D$ such that $|\operatorname{OE}_k(D)| \neq |\operatorname{EE}_k(D)|$. In this paper, we give another characterization of $k$-colourable graphs: a graph $G$ is $k$-colourable if and only if $G$ has an orientation $D$ such that $|\operatorname{OE}_{k-1}^s(D)| \neq |\operatorname{EE}_{k-1}^s(D)|$. We extend the characterizations of $k$-colourable graphs to $k$-colourable signed graphs: If $k$ is an even integer, then a signed graph $(G,\sigma)$ is $k$-colourable if and only if $G$ has an orientation $D$ such that $|\operatorname{OE}_k(D)| \neq |\operatorname{EE}_k(D)|$; if $k$ is an odd integer, then $(G,\sigma)$ is $k$-colourable if and only if $G$ has an orientation $D$ such that $|\operatorname{OE}_{k-1}^s(D)| \neq |\operatorname{EE}_{k-1}^s(D)|$, where a (special) modulo-$k$ Eulerian sub-digraph is even or odd if it has an even or odd number of positive edges. The characterization of $k$-colourable signed graphs for even $k$ (respectively, for odd $k$) fails for odd $k$ (respectively, for even $k$).

Extremal properties of flood-filling games

The problem of determining the number of flooding operations required to make a given coloured graph monochromatic in the one-player combinatorial game Flood-It has been studied extensively from an algorithmic point of view, but basic questions about the maximum number of moves that might be required in the worst case remain unanswered. We begin a systematic investigation of such questions, with the goal of determining, for a given graph, the maximum number of moves that may be required, taken over all possible colourings. We give several upper and lower bounds on this quantity for arbitrary graphs and show that all of the bounds are tight for trees; we also investigate how much the upper bounds can be improved if we restrict our attention to graphs with higher edge-density.