Colorable Graphs Research Articles

Decomposing a planar graph without triangular 4-cycles into a matching and a 3-colorable graph

Let c1,c2,…,ck be non-negative integers. A graph G is (c1,c2,…,ck)-colorable if the vertex set of G can be partitioned into k sets V1,V2,…,Vk, such that G[Vi], the subgraph induced by Vi, has maximum degree at most ci for i∈[k]. In this paper, we prove that every planar graph without triangular 4-cycles is (1,1,1)-colorable. Consequently, such planar graphs can be decomposed into a matching and a 3-colorable graph.

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 $$\max \left\{ 5,2\left\lceil \frac{t-1}{2}\right\rceil -2\right\} $$ many colors. If the input graph is triangle-free, we only need $$\max \left\{ 4,\left\lceil \frac{t-1}{2}\right\rceil +1\right\} $$ 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.

A Computer Approach to Determine the Densest Translative Tetrahedron Packings

In 1900, as a part of his 18th problem, Hilbert proposed the question to determine the densest congruent (or translative) packings of a given solid, such as the unit ball or the regular tetrahedron of unit edges. Up to now, our knowledge about this problem is still very limited, excepting the ball case. It is conjectured that, for some particular solids such as tetrahedra, cuboctahedra, and octahedra, their maximal translative packing densities and their maximal lattice packing densities are identical. To attack this conjecture, this article suggests a computer approach to determine the maximal local translative packing density of a given polytope, by studying associated color graphs and applying optimization. In particular, all the tetrahedral case, the cuboctahedral case and the octahedral case of the conjecture have been reduced into finite numbers of manageable optimization problems.

On local super h-decomposition antimagic total coloring of graphs

The writer introduces of local super H−decomposition antimagic total coloring of G. Let G(V, E) be a graph with vertex set V and edge set E. A bijection f : V(G) ∪ E(G) → {1, 2, 3, …, |V(G)| + |E(G)|} is called a local H−decomposition antimagic total labeling for any two adjacent subgraph H1 and H2, wt(H1) ≠ wt(H2), where wt(H) = ∑v∈ V(H)f(v) + ∑e∈ E(H)f(e). Thus, any local super H−decomposition antimagic total labeling induces a proper subgraph coloring of G if each subgraph H is assigned by color wt(H). A coloring graph of local super H-decomposition antimagic total coloring is an assignment of colors minimum number. The colors minimum number of graphs namely is chromatic number, denoted by γlaH(G). In this paper, the writer studies the local super H−decomposition antimagic total coloring of graphs namely path graph, cycle graph, prims graph, and jahangir graph.

Geometry of Generated Groups with Metrics Induced by Their Cayley Color Graphs

Abstract Let G be a group and let S be a generating set of G. In this article,we introduce a metric dC on G with respect to S, called the cardinal metric.We then compare geometric structures of (G, dC ) and (G, dW ), where dW denotes the word metric. In particular, we prove that if S is finite, then (G, dC ) and (G, dW ) are not quasiisometric in the case when (G, dW ) has infinite diameter and they are bi-Lipschitz equivalent otherwise. We also give an alternative description of cardinal metrics by using Cayley color graphs. It turns out that colorpermuting and color-preserving automorphisms of Cayley digraphs are isometries with respect to cardinal metrics.

Monochromatic Cycle Partitions of $2$-Coloured Graphs with Minimum Degree $3n/4$

Balogh, Barát, Gerbner, Gyárfás, and Sárközy made the following conjecture. Let $G$ be a graph on $n$ vertices with minimum degree at least $3n/4$. Then for every $2$-edge-colouring of $G$, the vertex set $V(G)$ may be partitioned into two vertex-disjoint cycles, one of each colour.
 We prove this conjecture for large $n$, improving approximate results by the aforementioned authors and by DeBiasio and Nelsen.

RGB Color Cluster and Graph Coloring Algorithm for Partial Color Blind Correction

Partial color blindness is an anomaly occurring to 5-8% of the world's population. Color correction using re-coloring algorithm usually can be used to help partial color blindness patient. One of the existing re-coloring techniques is to use the RGB color cluster technique and combine it by utilizing the brute force algorithm to perform color tracing for correcting the color. It has massive time and memory complexity. This research aims to create a correction technique for color blind people using RGB Color Cluster combined with Graph Coloring Algorithm. The first process is to get the RGB color cluster for color blind subject. After getting RGB color cluster from the subject then the image which want to be corrected is grouped based on RGB color cluster. The threshold of color grouping in image is done by utilizing the upper and lower bound values of the RGB color cluster. After the cluster is grouped, then we can represent neighbourhood between the colors by utilizing graph. The adjacent color group will be a neighbour. The next process is color re-coloring using graph coloring algorithm. In graph coloring algorithm, same color group is prohibited to become neighbour. In this research, graph coloring algorithm is used to prevent 2 colors that are look almost similar for become neighbours because it will cause the subject cannot distinguish it. Re-coloring is done by increasing and decreasing the color intensity of a set of colors. This technique succeeds in decreasing the complexity of the brute force algorithm from O(N4) to O(2N2) where the first N2 is the complexity of building the cluster group and the second N2 is the complexity of the re- coloring. In addition, the color of the object becomes more natural because Re-coloring is based on color group not pixel based.

Horizontal cooperation in a multimodal public transport system: The profit allocation problem

We analyse the situation involving several transport companies in an intermodal public transport system. It is assumed that they cooperate by offering travel cards which can be used on all available means of transport. In order to solve the problem of profit allocation among the companies involved, we first introduce a coloured graph which describes the transport network and then propose a game theoretical approach to share the profit among the companies. Two new allocation rules, called the Coloured Egalitarian Solution and the Coloured Cost Proportional Solution, are introduced and axiomatically characterised. Furthermore, it is shown that both solutions provide stable allocations in the sense that they cannot be improved upon by any coalition of players.

Planar and Non Planar Construction of - Uniquely Colorable Graph

A uniquely colorable graph G whose chromatic partition contains atleast one g - set is termed as a g - uniquely colorable graph. In this paper, we provide necessary and sufficient condition for and G* to be g - uniquely colorable whenever G g- uniquely colorable and also provide constructive characterization to show that whenever G is g- uniquely colorable such that |P | ³ 2, G can be both planarand non planar.

Tiling Tripartite Graphs with 3-Colorable Graphs: The Extreme Case

There is a sufficiently large $N\in h\mathbb{N}$ such that the following holds. If $G$ is a tripartite graph with $N$ vertices in each vertex class such that every vertex is adjacent to at least $2N/3+2h-1$ vertices in each of the other classes, then $G$ can be tiled perfectly by copies of $K_{h,h,h}$. This extends work by two of the authors [Electron. J. Combin, 16(1), 2009] and also gives a sufficient condition for tiling by any fixed 3-colorable graph. Furthermore, we show that $2N/3+2h-1$ in our result can not be replaced by $2N/3+ h-2$ and that if $N$ is divisible by $6h$, then we can replace it with the value $2N/3+h-1$ and this is tight.

Connectivity of Minimum Non-5-injectively Colorable Planar Cubic Graphs

Suppose that G is a planar cubic graph with $$\chi _i(G)>5$$. We show that if $$\chi _i(H)<\chi _i(G)$$ for each planar cubic graph H of order less than G, then G is either a 3-connected simple planar cubic graph, or a planar graph obtained from a simple cubic 3-connected planar graph by adding some earrings. This shows that a minimum non-5-injectively colorable simple planar cubic graph must be 3-connected.

Anesthesiology

This concise book is intended to be a comprehensive, easy-to-read text aimed both at anesthesiology trainees preparing for examinations and experienced practitioners needing a succinct reference guide. Authored by a number of experienced practitioners in the field of anesthesiology from across the United States, the editors have structured this textbook to describe common challenges in clinical anesthesia in a question–answer format. The book comprises 12 main, specialty-specific sections and a total of 60 chapters. Each chapter starts with a hypothetical clinical scenario relevant to a particular subspecialty in anesthesiology. A large number of chapters address the cardiovascular system and anesthetic considerations, for example, ischemic heart disease, valvular abnormalities, or uncommon conditions such as care of a patient with a transplanted heart presenting for noncardiac surgery. In every chapter, the starting clinical scenarios are well structured and provide a comprehensive account of the clinical issue, including relevant investigations. The main content of each chapter is in a format of a series of simple questions and succinct answers. Wherever indicated, basic understanding of anatomy and physiology is discussed, such as applied anatomy of coronary circulation, the denervated heart or nomenclature, and classification of pacemakers. The anesthetic implications in each of these situations are covered explicitly and referenced appropriately. The style of presentation is one of the main strengths of this book. Using a problem-based approach, the authors immediately direct readers to the matter at hand, that is, the clinical situation and how to safely provide anesthesia care while facilitating critical thinking and problem-solving skills. The questions guide readers through the logical steps in safe anesthetic management of various elective and emergency cases in great detail. Current evidence and guidelines are cited wherever necessary, such as the 2014 ACC/AHA Guidelines on Perioperative Cardiovascular Evaluation and Management of Patients Undergoing Noncardiac Surgery,1 or the Universal Definition of Myocardial infarction.2 Chapters on intraoperative coagulopathies, difficult airway management, and obstetric anesthesia also cover the applied pathophysiology and anesthetic implications quite well. The section on pediatric anesthesiology topics is comprehensive, and particularly, neonatal resuscitation and premature infant are described. Because this is the first edition of this book, it can be appraised using Yao and Artusio’s Anesthesiology: Problem-Oriented Patient Management,3 a core textbook, now in its 8th edition. Both books are laid out in a problem-based format using clinically relevant case scenarios. The table of contents is similar; both books deal with the topics in a well-organized questionanswer format. The Yao and Artusio book is a major reference text and covers each topic in extensive detail, with preoperative, intraoperative, and postoperative phases clearly yet compactly presented, referenced, and well illustrated where appropriate. The Anesthesiology case reviews book touches on the most relevant topics; however, the level of detail is not as extensive as that of the Yao and Artusio book. Some important and current practices in anesthesiology have been overlooked in this book, such as important considerations for use of newer reversal agents for nondepolarizing neuromuscular blockers (sugammadex) or use of gastric ultrasound for patients with a full stomach. Moreover, obstructive sleep apnea is not described as a condition in its own right, but as an association with obesity. The recent updated STOP-Bang questionnaire4 and the 2016 Society of Anesthesia and Sleep Medicine Guidelines on Preoperative Screening and Assessment of Adult Patients with Obstructive Sleep Apnea5 are also missing. In the regional anesthesia chapters, brachial plexus anatomy illustrations are presented in color; however, there is limited mention of the use of ultrasound for nerve blocks, a standard of care practiced in many institutions worldwide. The addition of colorful illustrations, graphs, and figures could make this book more readable. Some integral anesthesia topics such as acute and chronic pain management are mentioned very superficially, and some readers may find this an important exclusion of a significant knowledge area. Overall, this is a good book, serving as a handy resource for both trainees and clinicians who like to review high-yield topics in a case-based format. This book finds its place somewhere between being a useful concise pocket handbook for bedside reference, such as Oxford Handbook for Anesthesia,6 and a case-based reference book with more detail, such as the Yao and Artusio book.3 In the end, if a reader would like to use a textbook that touches on most of the relevant clinical topics in anesthesia or prepare for examinations, this book will serve as an important addition to their collection of useful resources. Najia Hasan, MBBS, FRCADepartment of Anaesthesia and Intensive CarePrince Charles HospitalCwm Taf University Health BoardWales, United Kingdom Mandeep Singh, MBBS, MD, MSc, FRCPCDepartment of Anesthesiology and Pain ManagementToronto Western HospitalUniversity Health NetworkUniversity of TorontoToronto, Ontario, CanadaDepartment of Anaesthesiology and Pain ManagementWomen’s College HospitalToronto, Ontario, CanadaToronto Sleep and Pulmonary CentreToronto, Ontario, Canada[email protected]

A Note on Not-4-List Colorable Planar Graphs

The Four Color Theorem states that every planar graph is properly 4-colorable. Moreover, it is well known that there are planar graphs that are non-$4$-list colorable. In this paper we investigate a problem combining proper colorings and list colorings. We ask whether the vertex set of every planar graph can be partitioned into two subsets where one subset induces a bipartite graph and the other subset induces a $2$-list colorable graph. We answer this question in the negative strengthening the result on non-$4$-list colorable planar graphs.

Defective 2-colorings of planar graphs without 4-cycles and 5-cycles

A 2-coloring is a coloring of vertices of a graph with colors 1 and 2. Define Vi≔{v∈V(G):c(v)=i} for i=1 and 2. We say that G is (d1,d2)-colorable if G has a 2-coloring such that Vi is an empty set or the induced subgraph G[Vi] has the maximum degree at most di for i=1 and 2. Let G be a planar graph without 4-cycles and 5-cycles. We show that the problem to determine whether G is (0,k)-colorable is NP-complete for every positive integer k. Moreover, we construct non-(1,k)-colorable planar graphs without 4-cycles and 5-cycles for every positive integer k. In contrast, we prove that G is (d1,d2)-colorable where (d1,d2)=(4,4),(3,5), and (2,9).

Word-Representable Graphs: a Survey

Letters x and y alternate in a word w if after deleting all letters but x and y in w we get either a word xyxy... or a word yxyx... (each of these words can be of odd or even length). A graph G = (V,E) is word-representable if there is a finite word w over an alphabet V such that the letters x and y alternate in w if and only if xy ∈ E. The word-representable graphs include many important graph classes, in particular, circle graphs, 3-colorable graphs and comparability graphs. In this paper we present the full survey of the available results on the theory of word-representable graphs and the most recent achievements in this field.

Connectivity and Hamiltonicity of Canonical Colouring Graphs of Bipartite and Complete Multipartite Graphs

A k-colouring of a graph G with colours 1 , 2 , … , k is canonical with respect to an ordering π = v 1 , v 2 , … , v n of the vertices of G if adjacent vertices are assigned different colours and, for 1 ≤ c ≤ k , whenever colour c is assigned to a vertex v i , each colour less than c has been assigned to a vertex that precedes v i in π . The canonical k-colouring graph of G with respect to π is the graph Can k π ( G ) with vertex set equal to the set of canonical k-colourings of G with respect to π , with two of these being adjacent if and only if they differ in the colour assigned to exactly one vertex. Connectivity and Hamiltonicity of canonical colouring graphs of bipartite and complete multipartite graphs is studied. It is shown that for complete multipartite graphs, and bipartite graphs there exists a vertex ordering π such that Can k π ( G ) is connected for large enough values of k. It is proved that a canonical colouring graph of a complete multipartite graph usually does not have a Hamilton cycle, and that there exists a vertex ordering π such that Can k π ( K m , n ) has a Hamilton path for all k ≥ 3 . The paper concludes with a detailed consideration of Can k π ( K 2 , 2 , … , 2 ) . For each k ≥ χ and all vertex orderings π , it is proved that Can k π ( K 2 , 2 , … , 2 ) is either disconnected or isomorphic to a particular tree.

On Uniquely k-List Colorable Planar Graphs, Graphs on Surfaces, and Regular Graphs

A graph G is called uniquely k -list colorable (UkLC) if there exists a list of colors on its vertices, say $$L=\lbrace S_v \mid v \in V(G) \rbrace $$ , each of size k, such that there is a unique proper list coloring of G from this list of colors. A graph G is said to have property M(k) if it is not uniquely k-list colorable. Mahmoodian and Mahdian (Ars Comb 51:295–305, 1999) characterized all graphs with property M(2). For $$k\ge 3$$ property M(k) has been studied only for multipartite graphs. Here we find bounds on M(k) for graphs embedded on surfaces, and obtain new results on planar graphs. We begin a general study of bounds on M(k) for regular graphs, as well as for graphs with varying list sizes.

Three colorability characterized by shrinking of locally connected subgraphs into triangles

We consider a recursive reduction of a graph where each 3-colorable locally connected subgraph is replaced by a triangle so that equally colored vertices coincide into one vertex. There is one-to-one correspondence between 3-colorings of the original graph and the graph resulting from the reduction. Using this approach we characterize classes of 3-colorable graphs and present polynomial time algorithms that recognize graphs from the classes and color them by three colors.

A new distribution-free model for disassembly line balancing problem with stochastic task processing times

Effective conduct with End of Life (EOL) products is a hot research topic in green and smart manufacturing. For EOL product recycling and remanufacturing, a fundamental problem is to design an efficient disassembly line under consideration of stochastic task processing times. This problem focuses on selecting alternative task processes, determining the number of opened workstations, and assigning operational tasks to the workstations. The goal is to minimise the total cost consisting of workstation operational cost and hazardous component processing cost. Most existing works assume that the probability distribution of task processing times can be estimated, however, it is often not likely to access the complete probability distribution due to various difficulties. Therefore, this study investigates disassembly line design with the assumption that only the mean, standard deviation and an upper bound of task processing times are known. Our main contributions include: (i) a new decomposition color graph is proposed to intuitively describe all possible processes, (ii) a new distribution-free model is proposed, and (iii) some problem properties are established to solve the model. Experimental results show that the distribution-free model can effectively deal with stochastic task processing times without given probability distributions.

A DNA Computing Model for the Graph Vertex Coloring Problem Based on a Probe Graph

The biggest bottleneck in DNA computing is exponential explosion, in which the DNA molecules used as data in information processing grow exponentially with an increase of problem size. To overcome this bottleneck and improve the processing speed, we propose a DNA computing model to solve the graph vertex coloring problem. The main points of the model are as follows: ① The exponential explosion problem is solved by dividing subgraphs, reducing the vertex colors without losing the solutions, and ordering the vertices in subgraphs; and ② the bio-operation times are reduced considerably by a designed parallel polymerase chain reaction (PCR) technology that dramatically improves the processing speed. In this article, a 3-colorable graph with 61 vertices is used to illustrate the capability of the DNA computing model. The experiment showed that not only are all the solutions of the graph found, but also more than 99% of false solutions are deleted when the initial solution space is constructed. The powerful computational capability of the model was based on specific reactions among the large number of nanoscale oligonucleotide strands. All these tiny strands are operated by DNA self-assembly and parallel PCR. After thousands of accurate PCR operations, the solutions were found by recognizing, splicing, and assembling. We also prove that the searching capability of this model is up to O(359). By means of an exhaustive search, it would take more than 896 000 years for an electronic computer (5 × 1014 s−1) to achieve this enormous task. This searching capability is the largest among both the electronic and non-electronic computers that have been developed since the DNA computing model was proposed by Adleman’s research group in 2002 (with a searching capability of O(220)).