Colorable Graphs Research Articles

Structural Representation of Colon Tissue using Delaunay Triangle for Cancer Detection and Grading

In human body, the cells are arranged in a particular pattern. Neoplastic diseases such as cancer may cause a change in these arrangements. Different methods can be used to quantify these patterns. This study mainly focuses on the structural representation of colon tissue and the graph features used to diagnose and grade the cancer. The different graph techniques used in structural method are color graphs, colorless graphs and probabilistic graphs. The inputs for the system are histopathological images. These images are pre-processed and then clustered according to luminal, stromal and nucleus components using k-means clustering algorithm. Then the centroids of each component are found out using region-props algorithm for each cluster in the whole image. Then these centroids are eroded and represented as disk structures and these disks are considered as set of nodes. Finally Delaunay diagram is created by connecting the nodes using different colored edges. From the diagram, set of features such as edge length, average edge length and diameter are extracted. These features are given to the ANN (Artificial Neural Network), which will accurately classify the images according to the grade of cancer to which they belong. General Terms Algorithms, Classification, Staging

Color graph based wavelet transform with perceptual information

In this paper, we propose a numerical strategy to define a multiscale analysis for color and multicomponent images based on the representation of data on a graph. Our approach consists in computing the graph of an image using the psychovisual information and analysing it by using the spectral graph wavelet transform. We suggest introducing color dimension into the computation of the weights of the graph and using the geodesic distance as a means of distance measurement. We thus have defined a wavelet transform based on a graph with perceptual information by using the CIELab color distance. This new representation is illustrated with denoising and inpainting applications. Overall, by introducing psychovisual information in the graph computation for the graph wavelet transform we obtain very promising results. Therefore results in image restoration highlight the interest of the appropriate use of color information.

Triangle-free Uniquely 3-Edge Colorable Cubic Graphs

This paper presents infinitely many new examples of triangle-free uniquely 3-edge colorable cubic graphs. The only such graph previously known was given by Tutte in 1976.

Online multi-coloring with advice

We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. For hexagonal graphs, negative results trivially carry over to 3-colorable graphs, and most of our positive results do as well. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time.

Continuous limit of discrete quantum walks

Quantum walks can be defined in two quite distinct ways: discrete-time and continuous-time quantum walks (DTQWs and CTQWs). For classical random walks, there is a natural sense in which continuous-time walks are a limit of discrete-time walks. Quantum mechanically, in the discrete-time case, an additional "coin space" must be appended for the walk to have nontrivial time evolution. Continuous-time quantum walks, however, have no such constraints. This means that there is no completely straightforward way to treat a CTQW as a limit of DTQW, as can be done in the classical case. Various approaches to this problem have been taken in the past. We give a construction for walks on $d$-regular, $d$-colorable graphs when the coin flip operator is Hermitian: from a standard DTQW we construct a family of discrete-time walks with a well-defined continuous-time limit on a related graph. One can think of this limit as a {\it coined} continuous-time walk. We show that these CTQWs share some properties with coined DTQWs. In particular, we look at spatial search by a DTQW over the 2-D torus (a grid with periodic boundary conditions) of size $\sqrt{N}\times\sqrt{N}$, where it was shown \nocite{AAmbainis08} that a coined DTQW can search in time $O(\sqrt{N}\log{N})$, but a standard CTQW \nocite{Childs2004} takes $\Omega(N)$ time to search for a marked element. The continuous limit of the DTQW search over the 2-D torus exhibits the $O(\sqrt{N}\log{N})$ scaling, like the coined walk it is derived from. We also look at the effects of graph symmetry on the limiting walk, and show that the properties are similar to those of the DTQW as shown in \cite{HariKrovi2007}.

Hadwiger’s Conjecture for inflations of 3-chromatic graphs

Hadwiger’s Conjecture states that every k-chromatic graph has a complete minor of order k. A graph G′ is an inflation of a graph G if G′ is obtained from G by replacing each vertex v of G by a clique Cv and joining two vertices of distinct cliques by an edge if and only if the corresponding vertices of G are adjacent. We present an algorithm for computing an upper bound on the chromatic number χ(G′) of any inflation G′ of any 3-chromatic graph G. As a consequence, we deduce that Hadwiger’s Conjecture holds for any inflation of any 3-colorable graph.

Every 3-colorable graph has a faithful representation in the odd-distance graph

We prove that every finite 3-colorable graph has an odd-distance faithful representation in the plane. In other words, we can draw it in the plane so that any two vertices are connected by an edge if and only if their distance is an odd integer.

Berge–Fulkerson Conjecture on Certain Snarks

A snark is a connected, bridgeless cubic graph with chromatic index equal to 4. The Berge–Fulkerson conjecture proposed in 1971 states that every bridgeless cubic graph contains a family of six perfect matchings such that each edge is contained in exactly two of them. This conjecture holds trivialy for 3-edge colorable graphs. Thus a possible minimum counterexample for the conjecture is a snark. In this paper we shown that the conjecture holds for families of snarks such as Loupekhine snarks of first and second kind and the Watkins snark. We also determine the excessive index for Loupekhine snarks of first and second kind.

Cycles Avoiding a Color in Colorful Graphs

AbstractThe Ramsey numbers of cycles imply that every 2‐edge‐colored complete graph on n vertices contains monochromatic cycles of all lengths between 4 and at least . We generalize this result to colors by showing that every k‐edge‐colored complete graph on vertices contains ‐edge‐colored cycles of all lengths between 3 and at least .

On an interval (1, 1)-coloring of incidentors of interval colorable graphs

A graph is called interval colorable if there exists its proper edge coloring such that for every vertex the set of colors used for coloring the edges incident with the vertex forms an interval. A subdivision of a graph is a graph obtained by replacing each edge with a path of length 2. Petrosyan and Khachatryan posed a conjecture that the subdivision of every interval colorable graph is interval colorable. In this paper we prove this conjecture.

Exact Algorithms for Intervalizing Coloured Graphs

In the Intervalizing Coloured Graphs problem, one must decide for a given graph G = (V, E) with a proper vertex colouring of G whether G is the subgraph of a properly coloured interval graph. For the case that the number of colors is fixed, we give an exact algorithm that uses 2O(n/logn)$2^{\mathcal {O}(n/\log n)}$ time. We also give an O?(2n)$\mathcal {O}^{\ast }(2^{n})$ algorithm for the case that the number of colors is not fixed.

Near-Colorings: Non-Colorable Graphs and NP-Completeness

A graph $G$ is $(d_1,...,d_l)$-colorable if the vertex set of $G$ can be partitioned into subsets $V_1,\ldots ,V_l$ such that the graph $G[V_i]$ induced by the vertices of $V_i$ has maximum degree at most $d_i$ for all $1 \leq i \leq l$. In this paper, we focus on complexity aspects of such colorings when $l=2,3$. More precisely, we prove that, for any fixed integers $k,j,g$ with $(k,j) \neq (0,0)$ and $g\geq3$, either every planar graph with girth at least $g$ is $(k,j)$-colorable or it is NP-complete to determine whether a planar graph with girth at least $g$ is $(k,j)$-colorable. Also, for any fixed integer $k$, it is NP-complete to determine whether a planar graph that is either $(0,0,0)$-colorable or non-$(k,k,1)$-colorable is $(0,0,0)$-colorable. Additionally, we exhibit non-$(3,1)$-colorable planar graphs with girth 5 and non-$(2,0)$-colorable planar graphs with girth 7.

Colouring graph by the kernel P system

The graph colouring problem is one of the most well-known NP-hard problems. It consists of dividing the set of nodes into classes, so that the nodes of each edge are not in the same class and the number of classes is the minimum corresponding to the number of colour classes. In this paper, we have used the kernel P systems to solve the graph colouring problem. To test the proposed system, three running examples are given using MeCoSim simulator.

The UCSC Genome Browser database: 2015 update.

Launched in 2001 to showcase the draft human genome assembly, the UCSC Genome Browser database (http://genome.ucsc.edu) and associated tools continue to grow, providing a comprehensive resource of genome assemblies and annotations to scientists and students worldwide. Highlights of the past year include the release of a browser for the first new human genome reference assembly in 4 years in December 2013 (GRCh38, UCSC hg38), a watershed comparative genomics annotation (100-species multiple alignment and conservation) and a novel distribution mechanism for the browser (GBiB: Genome Browser in a Box). We created browsers for new species (Chinese hamster, elephant shark, minke whale), ‘mined the web’ for DNA sequences and expanded the browser display with stacked color graphs and region highlighting. As our user community increasingly adopts the UCSC track hub and assembly hub representations for sharing large-scale genomic annotation data sets and genome sequencing projects, our menu of public data hubs has tripled.

New results on word-representable graphs

A graph G=(V,E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if (x,y)∈E for each x≠y. The set of word-representable graphs generalizes several important and well-studied graph families, such as circle graphs, comparability graphs, 3-colorable graphs, graphs of vertex degree at most 3, etc. By answering an open question from Halldórsson et al. (2011), in the present paper we show that not all graphs of vertex degree at most 4 are word-representable. Combining this result with some previously known facts, we derive that the number of n-vertex word-representable graphs is 2n23+o(n2).

Numerical Verification of a Full-field Deformation Analysis of a Specimen Loaded by Combined Loading

The contribution deals with an experimental investigation of deformation fields of a specimen loaded by combined loading. The experiment was done using non-contact optical method of low-speed digital image correlation. This technique allows investigate displacement as well strain fields. The experimentally obtained results depicted in the paper in a form of color maps and graphs were verified by a numerical simulation performed in Ansys, where two types of meshing were used to demonstrate the coherence between experiment and simulation.

Cayley sum color and anti-circulant graphs

Let G be a finite group and α:G→R be a real-valued function on G. The Cayley sum color graph Cay+(G,α) is a complete directed graph with vertex set G where each arc (x,y)∈G×G is associated with a color α(xy). If α is the characteristic function on a subset S of G, then the Cayley sum graph Cay+(G,S) is obtained. The anti-circulant matrix associated to a vector v is an n×n matrix whose rows are given by iterations of the anti-shift operator acting on v. We note that a graph is a Cayley sum graph of a cyclic group if and only if it is an anti-circulant graph, a graph whose adjacency matrix is anti-circulant. In this paper, we obtain some results on the isomorphisms, connectivity and vertex transitivity about anti-circulant graphs. We find the spectrum of Cayley sum color graphs of abelian groups and as a result we compute the spectrum of real anti-circulant matrices.

On the simultaneous edge coloring of graphs

A μ-simultaneous edge coloring of graph G is a set of μ proper edge colorings of G with a same color set such that for each vertex, the sets of colors appearing on the edges incident to that vertex are the same in each coloring and no edge receives the same color in any two colorings. The μ-simultaneous edge coloring of bipartite graphs has a close relation with μ-way Latin trades. Mahdian et al. (2000) conjectured that every bridgeless bipartite graph is 2-simultaneous edge colorable. Luo et al. (2004) showed that every bipartite graphic sequence S with all its elements greater than one, has a realization that admits a 2-simultaneous edge coloring. In this paper, the μ-simultaneous edge coloring of graphs is studied. Moreover, the properties of the extremal counterexample to the above conjecture are investigated. Also, a relation between 2-simultaneous edge coloring of a graph and a cycle double cover with certain properties is shown and using this relation, some results about 2-simultaneous edge colorable graphs are obtained.

A combinatorial approach to quantum error correcting codes

Motivated from the theory of quantum error correcting codes, we investigate a combinatorial problem that involves a symmetric n-vertices colorable graph and a group of operations (coloring rules) on the graph: find the minimum sequence of operations that maps between two given graph colorings. We provide an explicit algorithm for computing the solution of our problem, which in turn is directly related to computing the distance (performance) of an underlying quantum error correcting code. Computing the distance of a quantum code is a highly non-trivial problem and our method may be of use in the construction of better codes.

Note on property M(k) of some complete multipartite graphs

The property M(k) is a concept associated with the unique list coloring. A graph G has the property M(k) if for any collection of lists assigned to its vertices, each of size k, either there is no list coloring for G or there are at least two k-list colorings. The existing research results indicate that K1,1,1,r and K1⁎r,3 have the property M(3), and in addition K1⁎5,r and K1⁎r,5 have the property M(4) for every r≥2. The results above are extended to every k in this paper. We will show that for every r≥1, k≥2, K1⁎r,(2k−3) and K1⁎(2k−3),r have the property M(k). The conclusion will pave the way to characterizing unique k-list colorable complete multipartite graphs.