Colorable Graphs Research Articles

Mathematical Models to Predict and Analyze the Energy Consumption of a Domestic Refrigerator for Different Position of the Shelves

In this paper, a domestic refrigerator was tested to determine the power consumption for different positions of the shelves. The main contribution of this paper is the application of several strategies to analyze and model the power consumption of a domestic refrigerator when the fresh food shelves are changed of position. First, computer simulations were performed to analyze the power consumption using the moving average, the numerical derivative, and the Fourier transform. These simulations were used to study changes, periodic behaviors, minimum, maximum, and average values for the power consumption. Second, prediction methods were used to model the power consumption, these methods include cubic splines, the bilinear method, and artificial neural networks. With these methods, 2-D color graphs were built to predict the power consumption for any shelves positions. The validation results revealed that the cubic spline method provides the best results. Finally, it can be concluded that the models proposed in this paper provides new insights that can be used to design the internal compartments of the fresh food in order to try reducing the energy consumption.

3-Colorable Subclasses of $P_8$-Free Graphs

In this paper, we study 3-colorable graphs having no induced 8-vertex path and no induced cycles of specific lengths. We prove a characterization by critical graphs in three particular cases.

The Relation between Centre Folding of Graph and its Chromatic Number

Coloring graph is a good branch in graph theory and has a lot of application in our life. So, in this paper we have defined a new type of folding. Also, we have found strong relations between this folding of graph and its chromatic number. Theorems governing the relations to be obtained. Traffic lights problem here was solved.

Planar graph characterization of γ - Uniquely colorable graphs

A graph G = ( V, E ) is uniquely colorable if the chromatic number χ ( G ) = n and every n – coloring of G induces the same partition of V. Uniquely colorable graphs whose chromatic partition contains at least one γ - set is γ - uniquely colorable graphs In this paper, we characterize the planarity of γ - uniquely colorable graphs.

γ - Uniquely colorable graphs

A graph G = ( V, E ) is uniquely colorable if the chromatic number χ ( G ) = n and every n – coloring of G induces the same partition of V. In this paper, we introduce a new kind of graph called γ - uniquely colorable graphs. We obtain a necessary and sufficient condition for a graph to be γ - uniquely colorable graphs. We provide a constructive characterization of γ - uniquely colorable trees.

Minimum sum coloring for large graphs with extraction and backward expansion search

The Minimum Sum Coloring Problem (MSCP) is a relevant model tightly related to the classical vertex coloring problem (VCP). MSCP is known to be NP-hard, thus solving the problem for large graphs is particular challenging. Based on the general “reduce-and-solve” principle and inspired by the work for the VCP, we present an extraction and backward expansion search approach (EBES) to compute the upper and lower bounds for the MSCP on large graphs. The extraction phase reduces the given graph by extracting large collections of pairwise disjoint large independent sets (or color classes). The backward extension phase adds the extracted independent sets to recover the intermediate graphs while optimizing the sum coloring of each intermediate graph. We assess the proposed approach on a set of 35 large benchmark graphs with 450–4000 vertices from the DIMACS and COLOR graph coloring competitions. Computational results show that EBES is able to find improved upper bounds for 19 graphs and improved lower bounds for 12 graphs.

3-coloring triangle-free planar graphs with a precolored 9-cycle

Given a triangle-free planar graph G and a 9-cycle C in G, we characterize situations where a 3-coloring of C does not extend to a proper 3-coloring of G. This extends previous results when C is a cycle of length at most 8.

Graph-based video sequence matching using dominant colour graph profile (DCGP)

This paper presents a fast and effective technique for videos’ visual similarity detection and measurement using compact fixed-length signatures. The proposed technique (dominant colour graph profile DCGP) extracts and encodes the spatio-temporal information of a given video shot into a graph-based structure (tree) that fully captures this vital information. The graph structured properties are utilized to construct a fixed-length video signature of 112 decimal values per video shot. The encoded spatio-temporal information is extracted following channelling each video frame into a block-based structure, where the positions of respective blocks are tracked across video frames and encoded into multiple DCGP trees. The proposed technique provides a high matching speed (>2000 fps) and robust retrieval performance. The experiments on various standard and challenging datasets shows the framework’s robust performance, in terms of both, retrieval and computational performances.

Universal Completability, Least Eigenvalue Frameworks, and Vector Colorings

An embedding i mapsto p_iin mathbb {R}^d of the vertices of a graph G is called universally completable if the following holds: For any other embedding imapsto q_i~in mathbb {R}^{k} satisfying q_i^{T}q_j = p_i^{T}p_j for i = j and i adjacent to j, there exists an isometry mapping q_i to p_i for all iin V(G). The notion of universal completability was introduced recently due to its relevance to the positive semidefinite matrix completion problem. In this work we focus on graph embeddings constructed using the eigenvectors of the least eigenvalue of the adjacency matrix of G, which we call least eigenvalue frameworks. We identify two necessary and sufficient conditions for such frameworks to be universally completable. Our conditions also allow us to give algorithms for determining whether a least eigenvalue framework is universally completable. Furthermore, our computations for Cayley graphs on mathbb {Z}_2^n (n le 5) show that almost all of these graphs have universally completable least eigenvalue frameworks. In the second part of this work we study uniquely vector colorable (UVC) graphs, i.e., graphs for which the semidefinite program corresponding to the Lovász theta number (of the complementary graph) admits a unique optimal solution. We identify a sufficient condition for showing that a graph is UVC based on the universal completability of an associated framework. This allows us to prove that Kneser and q-Kneser graphs are UVC. Lastly, we show that least eigenvalue frameworks of 1-walk-regular graphs always provide optimal vector colorings and furthermore, we are able to characterize all optimal vector colorings of such graphs. In particular, we give a necessary and sufficient condition for a 1-walk-regular graph to be uniquely vector colorable.

Formula omitted]-WORM colorings: Results for [formula omitted]-connected graphs

Given two graphs F and G, an F-WORM coloring of G is an assignment of colors to its vertices in such a way that no F-subgraph of G is monochromatic or rainbow. If G has at least one such coloring, then it is called F-WORM colorable and W−(G,F) denotes the minimum possible number of colors. Here, we consider F-WORM colorings with a fixed 2-connected graph F and prove the following three main results: (1) For every natural number k, there exists a graph G which is F-WORM colorable and W−(G,F)=k; (2) It is NP-complete to decide whether a graph is F-WORM colorable; (3) For each k≥|V(F)|−1, it is NP-complete to decide whether a graph G satisfies W−(G,F)≤k. This remains valid on the class of F-WORM colorable graphs of bounded maximum degree. We also prove that for each n≥3, there exists a graph G and integers r and s such that s≥r+2, G has Kn-WORM colorings with exactly r and also with s colors, but it admits no Kn-WORM colorings with exactly r+1,…,s−1 colors. Moreover, the difference s−r can be arbitrarily large.

Incidence coloring of graphs with high maximum average degree

An incidence of an undirected graph G is a pair (v,e) where v is a vertex of G and e an edge of G incident with v. Two incidences (v,e) and (w,f) are adjacent if one of the following holds: (i) v=w, (ii) e=f or (iii) vw=e or f. An incidence coloring of G assigns a color to each incidence of G in such a way that adjacent incidences get distinct colors. In 2005, Hosseini Dolama and Sopena (2005) proved that every graph with maximum average degree strictly less than 3 can be incidence colored with Δ+3 colors. Recently, Bonamy et al. (2014) proved that every graph with maximum degree at least 4 and with maximum average degree strictly less than 73 admits an incidence (Δ+1)-coloring. In this paper we give bounds for the number of colors needed to color graphs having maximum average degrees bounded by different values between 4 and 6. In particular we prove that every graph with maximum degree at least 7 and with maximum average degree less than 4 admits an incidence (Δ+3)-coloring. This result implies that every triangle-free planar graph with maximum degree at least 7 is incidence (Δ+3)-colorable. We also prove that every graph with maximum average degree less than 6 admits an incidence (Δ+7)-coloring. More generally, we prove that Δ+k−1 colors are enough when the maximum average degree is less than k and the maximum degree is sufficiently large.

Approximability of Clique Transversal in Perfect Graphs

Given an undirected simple graph G, a set of vertices is an r-clique transversal if it has at least one vertex from every r-clique. Such sets generalize vertex covers as a vertex cover is a 2-clique transversal. Perfect graphs are a well-studied class of graphs on which a minimum weight vertex cover can be obtained in polynomial time. Further, an r-clique transversal in a perfect graph is also a set of vertices whose deletion results in an $$(r-1)$$ -colorable graph. In this work, we study the problem of finding a minimum weight r-clique transversal in a perfect graph. This problem is known to be $$\mathsf {NP}$$ -hard for $$r \ge 3$$ and admits a straightforward r-approximation algorithm. We describe two different $$\frac{r+1}{2}$$ -approximation algorithms for the problem. Both the algorithms are based on (different) linear programming relaxations. The first algorithm employs the primal–dual method while the second uses rounding based on a threshold value. We also show that the problem is APX-hard and describe hardness results in the context of parameterized algorithms and kernelization.

Strengthening topological colorful results for graphs

Various results ensure the existence of large complete and colorful bipartite graphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied. We generalize three theorems of this kind, respectively due to Simonyi and Tardos 2006), Simonyi et al. (2013), and Chen 2011). As a consequence of the generalization of Chen’s theorem, we get new families of graphs whose chromatic number equals their circular chromatic number and that satisfy Hedetniemi’s conjecture for the circular chromatic number.

ON CERTAIN HYPERPLANE ARRANGEMENTS AND COLORED GRAPHS

We exhibit a one-to-one correspondence between 3-colored graphs and subarrangements of certain hyperplane arrangements denoted <TEX>${\mathcal{J}}_n$</TEX>, <TEX>$n{\in}{\mathbb{N}}$</TEX>. We define the notion of centrality of 3-colored graphs, which corresponds to the centrality of hyperplane arrangements. Via the correspondence, the characteristic polynomial <TEX>${\chi}{\mathcal{J}}_n$</TEX> of <TEX>${\mathcal{J}}_n$</TEX> can be expressed in terms of the number of central 3-colored graphs, and we compute <TEX>${\chi}{\mathcal{J}}_n$</TEX> for n = 2, 3.

Heat transfer modeling and temperature experiments of crystalline silicon photovoltaic modules

In this study, the time-dependent thermal performance of crystalline silicon photovoltaic (PV) modules with glass-glass (GG) and glass-back sheet (GB) configurations were investigated. A heat transfer model for PV modules was created in COMSOL Multiphysics environment, to illustrate the time dependent thermal behavior of the implemented module during operation. A thorough understanding of the heat sources, temperature distribution, as well as the performance of heat dispersion and temperature uniform by the solar module in operation was presented with colorful maps, graphs and temperature difference between different locations within PV module. All the analysis leading to the conclusions that the GG modules has a better temperature uniform and heat dispersion performance with respect to GB module although GG module undertakes a slightly higher temperature. However, the temperature of GG module can also be reduced by choosing proper encapsulate and protective covers.

Spectrum, distance spectrum, and Wiener index of wreath products of complete graphs

We describe the adjacency matrix and the distance matrix of the wreath product of two complete graphs, and we give an explicit computation of their spectra. As an application, we deduce the spectrum of the transition matrix of the Lamplighter random walk over a complete base graph, with a complete color graph. Finally, an explicit computation of the Wiener index is given.

Coloring 3-Colorable Graphs with Less than n 1/5 Colors

We consider the problem of coloring a 3-colorable graph in polynomial time using as few colors as possible. We first present a new combinatorial algorithm using Õ ( n 4/11 ) colors. This is the first combinatorial improvement since Blum’s Õ ( n 3/8 ) bound from FOCS’90. Like Blum’s algorithm, our new algorithm composes immediately with recent semi-definite programming approaches, and improves the best bound for the polynomial time algorithm for the coloring of 3-colorable graphs from O ( n 0.2072 ) colors by Chlamtac from FOCS’07 to O ( n 0.2049 ) colors. Next, we develop a new recursion tailored for combination with semi-definite approaches, bringing us further down to O ( n 0.19996 ) colors.

Realization of Aggregate Applications using Dynamic Behaviour of Cellular Automata

Cellular automata have proved many of its capabilities and have bestowed a lot in many fields. With the emergence of CA, fabric pattern production has increased in less amount of time. For weaving, cellular automata start with some pattern, then continues with a sequence of steps to produce a new pattern or to change the colour of the cell in a lattice, by using particular transition rules. The main purpose of using cellular automata algorithm as it provides the superlative edge maps and the outstanding quality with one pixel wide edge, with edges having no breaks. We assessed cellular automata capabilities and CA a lot contributions in pattern generation, transformation, and data processing. This research paper reflects the integration of cellular automata across different disciplines. In this paper, we assess the computationally enriched CA rules facilitating spot detection in the study of medical images for cancer diagnosis. The dynamic behaviour of CA increases the scope of transformation and makes it practical for morphing. CA provides prediction of protein structural class and processes dynamic simulation of protein. Flexibility of CA facilitates parallel processing using VLSI, colour graph modelling the linear rules, and its exercise in fabric weaving.

Forbidden structures for planar perfect consecutively colourable graphs

A consecutive colouring of a graph is a proper edge colouring with natural numbers in which the colours of edges incident with each vertex form an interval of integers. The idea of this colouring was introduced in 1987 by Asratian and Kamalian [1] for bipartite graphs and developed for all graphs by Giaro and Kubale. Sevastjanov showed that the corresponding decision problem is NP -complete even restricted to the class of bipartite graphs. We focus our attention on the class of consecutively colourable graphs whose all induced subgraphs are consecutively colourable, too. We call elements of this class perfect consecutively colourable to emphasise the conceptual similarity to perfect graphs. Obviously, the class of perfect consecutively colourable graphs is induced hereditary, so it can be characterized by the family of induced forbidden graphs. In this work we give a necessary and sufficient conditions that must be satisfied by the Rosette of Sevastjanov to be a forbidden graph for the class of perfect consecutively colourable graphs. Along the way, we show the exact values of the deficiency of all Rosettes of Sevastjanov, which improves the earlier known estimating result. It should be mentioned that the deficiency of a graph measures its closeness to the class of consecutively colourable graphs. Finally, we motivate the investigation of graphs considered here by showing their connection to the class of planar perfect consecutively colourable graphs.

A Novel Edge Cover based Graph Coloring Algorithm

Graph Colouring Problem is a well-known NP-Hard problem. In Graph Colouring Problem (GCP) all vertices of any graph must be coloured in such a way that no two adjacent vertices are coloured with the same colour. In this paper, a new algorithm is proposed to solve the GCP. Proposed algorithm is based on finding vertex sets using edge cover method. In this paper implementation prospective of the algorithm is also discussed. Implemented algorithm is tested on various graph instances of DIMACS standards dataset. Algorithm execution time and a number of colours required to colour graph are compared with some other well-known Graph Colouring Algorithms. Variation in time complexity with reference to increasing in the number of vertices, a number of edges and an average degree of a graph are also discussed in this paper.