Search results for: Bond graph modeling

Authors: J. Baskar Babujee, J. Senbagamalar,

Abstract:

The topological distance between a pair of vertices i and j, which is denoted by d(vi, vj), is the number of edges of the shortest path joining i and j. The Wiener index W(G) is the sum of distances between all pairs of vertices of a graph G. W(G) = i

Keywords: Graph, Degree, Distance, Pendent vertex, Wiener index, Tree.

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Authors: Jean-Christophe Sibel, Sylvain Reynal, David Declercq

Abstract:

In the context of channel coding, the Generalized Belief Propagation (GBP) is an iterative algorithm used to recover the transmission bits sent through a noisy channel. To ensure a reliable transmission, we apply a map on the bits, that is called a code. This code induces artificial correlations between the bits to send, and it can be modeled by a graph whose nodes are the bits and the edges are the correlations. This graph, called Tanner graph, is used for most of the decoding algorithms like Belief Propagation or Gallager-B. The GBP is based on a non unic transformation of the Tanner graph into a so called region-graph. A clear advantage of the GBP over the other algorithms is the freedom in the construction of this graph. In this article, we explain a particular construction for specific graph topologies that involves relevant performance of the GBP. Moreover, we investigate the behavior of the GBP considered as a dynamic system in order to understand the way it evolves in terms of the time and in terms of the noise power of the channel. To this end we make use of classical measures and we introduce a new measure called the hyperspheres method that enables to know the size of the attractors.

Keywords: iterative decoder, LDPC, region-graph, chaos.

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Authors: Stefan Wagenpfeil, Felix Engel, Paul McKevitt, Matthias Hemmje

Abstract:

Multimedia Indexing and Retrieval is generally de-signed and implemented by employing feature graphs. These graphs typically contain a significant number of nodes and edges to reflect the level of detail in feature detection. A higher level of detail increases the effectiveness of the results but also leads to more complex graph structures. However, graph-traversal-based algorithms for similarity are quite inefficient and computation intensive, espe-cially for large data structures. To deliver fast and effective retrieval, an efficient similarity algorithm, particularly for large graphs, is mandatory. Hence, in this paper, we define a graph-projection into a 2D space (Graph Code) as well as the corresponding algorithms for indexing and retrieval. We show that calculations in this space can be performed more efficiently than graph-traversals due to a simpler processing model and a high level of parallelisation. In consequence, we prove that the effectiveness of retrieval also increases substantially, as Graph Codes facilitate more levels of detail in feature fusion. Thus, Graph Codes provide a significant increase in efficiency and effectiveness (especially for Multimedia indexing and retrieval) and can be applied to images, videos, audio, and text information.

Keywords: indexing, retrieval, multimedia, graph code, graph algorithm

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Authors: T. N. Janakiraman, J. Janet Lourds Rani

Abstract:

In practice, wireless networks has the property that the signal strength attenuates with respect to the distance from the base station, it could be better if the nodes at two hop away are considered for better quality of service. In this paper, we propose a procedure to identify delay preserving substructures for a given wireless ad-hoc network using a new graph operation G 2 – E (G) = G* (Edge difference of square graph of a given graph and the original graph). This operation helps to analyze some induced substructures, which preserve delay in communication among them. This operation G* on a given graph will induce a graph, in which 1- hop neighbors of any node are at 2-hop distance in the original network. In this paper, we also identify some delay preserving substructures in G*, which are (i) set of all nodes, which are mutually at 2-hop distance in G that will form a clique in G*, (ii) set of nodes which forms an odd cycle C2k+1 in G, will form an odd cycle in G* and the set of nodes which form a even cycle C2k in G that will form two disjoint companion cycles ( of same parity odd/even) of length k in G*, (iii) every path of length 2k+1 or 2k in G will induce two disjoint paths of length k in G*, and (iv) set of nodes in G*, which induces a maximal connected sub graph with radius 1 (which identifies a substructure with radius equal 2 and diameter at most 4 in G). The above delay preserving sub structures will behave as good clusters in the original network.

Keywords: Clique, cycles, delay preserving substructures, maximal connected sub graph.

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Authors: Saipunidzam Mahamad, Mohammad Noor Ibrahim, Rozana Kasbon, Chap Samol

Abstract:

Animated graph gives some good impressions in presenting information. However, not many people are able to produce it because the process of generating an animated graph requires some technical skills. This work presents Content Management System with Animated Graph (CMS-AG). It is a webbased system enabling users to produce an effective and interactive graphical report in a short time period. It allows for three levels of user authentication, provides update profile, account management, template management, graph management, and track changes. The system development applies incremental development approach, object-oriented concepts and Web programming technologies. The design architecture promotes new technology of reporting. It also helps user cut off unnecessary expenses, save time and learn new things on different levels of users. In this paper, the developed system is described.

Keywords: Animated Graph, Content Management System.

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Authors: Jing Lu, Malcolm Keech, Weiru Chen

Abstract:

Web usage mining is an interesting application of data mining which provides insight into customer behaviour on the Internet. An important technique to discover user access and navigation trails is based on sequential patterns mining. One of the key challenges for web access patterns mining is tackling the problem of mining richly structured patterns. This paper proposes a novel model called Web Access Patterns Graph (WAP-Graph) to represent all of the access patterns from web mining graphically. WAP-Graph also motivates the search for new structural relation patterns, i.e. Concurrent Access Patterns (CAP), to identify and predict more complex web page requests. Corresponding CAP mining and modelling methods are proposed and shown to be effective in the search for and representation of concurrency between access patterns on the web. From experiments conducted on large-scale synthetic sequence data as well as real web access data, it is demonstrated that CAP mining provides a powerful method for structural knowledge discovery, which can be visualised through the CAP-Graph model.

Keywords: concurrent access patterns (CAP), CAP mining and modelling, CAP-Graph, web access patterns (WAP), WAP-Graph, Web usage mining.

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Authors: Siong W. Lee, Kang H. Tan, En H. Yang

Abstract:

This paper presents the results of an experimental study undertaken to evaluate the local bond stress-slip response of short embedment of reinforcing bars in normal concrete (NC) and high performance fiber reinforced cement composites (HPFRCC) blocks. Long embedment was investigated as well to gain insights on the distribution of strain, slip, bar stress and bond stress along the bar especially in post-yield range. A total of 12 specimens were tested, by means of pull-out of the reinforcing bars from concrete blocks. It was found that the enhancement of local bond strength can be reached up to 50% and ductility of the bond behavior was improved significantly if HPFRCC is used. Also, under a constant strain at loaded end, HPFRCC has delayed yielding of bars at other location from the loaded end. Hence, the reduction of bond stress was slower for HPFRCC in comparison with NC. Due to the same reason, the total slips at loaded end for HPFRCC was smaller than NC as expected. Test results indicated that HPFRCC has better bond slip behavior which makes it a suitable material to be employed in anchorage zone such as beam-column joints.

Keywords: Bond stress, high performance fiber reinforced cement composites, slip, strain.

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Authors: Maruful H. Mazumder, Raymond I. Gilbert, Zhen- T. Chang

Abstract:

A reliable estimate of the average bond stress within the anchorage of steel reinforcing bars in tension is critically important for the design of reinforced concrete member. This paper describes part of a recently completed experimental research program in the Centre for Infrastructure Engineering and Safety (CIES) at the University of New South Wales, Sydney, Australia aimed at assessing the effects of different factors on the anchorage requirements of modern high strength steel reinforcing bars. The study found that an increase in the anchorage length and bar diameter generally leads to a reduction of the average ultimate bond stress. By the extension of a well established analytical model of bond and anchorage, it is shown here that the differences in the average ultimate bond stress for different anchorage lengths is associated with the variable degree of plastic deformation in the tensile zone of the concrete surrounding the bar.

Keywords: Anchorage, Bond stress, Development length, Reinforced concrete.

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Authors: Sizhong Zhou, Bingyuan Pu

Abstract:

Let G be a Hamiltonian graph. A factor F of G is called a Hamiltonian factor if F contains a Hamiltonian cycle. In this paper, two sufficient conditions are given, which are two neighborhood conditions for a Hamiltonian graph G to have a Hamiltonian factor.

Keywords: graph, neighborhood, factor, Hamiltonian factor.

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Authors: Shin-Shin Kao, Yuan-Kang Shih, Hsun Su

Abstract:

In this paper, we show that the conjecture of Chv tal, which states that any 1-tough graph is either a Hamiltonian graph or its complement contains a specific graph denoted by F, does not hold in general. More precisely, it is true only for graphs with six or seven vertices, and is false for graphs with eight or more vertices. A theorem is derived as a correction for the conjecture.

Keywords: Complement, degree sum, Hamiltonian, tough.

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Authors: Maryam Hsaini, Mostafa Nouri-Baygi

Abstract:

A critical problem in wireless sensor networks is limited battery and memory of nodes. Therefore, each node in the network could maintain only a subset of its neighbors to communicate with. This will increase the battery usage in the network because each packet should take more hops to reach its destination. In order to tackle these problems, spanner graphs are defined. Since each node has a small degree in a spanner graph and the distance in the graph is not much greater than its actual geographical distance, spanner graphs are suitable candidates to be used for the topology of a wireless sensor network. In this paper, we study Yao graphs and their behavior for a randomly selected set of points. We generate several random point sets and compare the properties of their Yao graphs with the complete graph. Based on our data sets, we obtain several charts demonstrating how Yao graphs behave for a set of randomly chosen point set. As the results show, the stretch factor of a Yao graph follows a normal distribution. Furthermore, the stretch factor is in average far less than the worst case stretch factor proved for Yao graphs in previous results. Furthermore, we use Yao graph for a realistic point set and study its stretch factor in real world.

Keywords: Wireless sensor network, spanner graph, Yao Graph.

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Authors: Kazuaki Maeda

Abstract:

This paper describes a code clone visualization method, called FC graph, and the implementation issues. Code clone detection tools usually show the results in a textual representation. If the results are large, it makes a problem to software maintainers with understanding them. One of the approaches to overcome the situation is visualization of code clone detection results. A scatter plot is a popular approach to the visualization. However, it represents only one-to-one correspondence and it is difficult to find correspondence of code clones over multiple files. FC graph represents correspondence among files, code clones and packages in Java. All nodes in FC graph are positioned using force-directed graph layout, which is dynami- cally calculated to adjust the distances of nodes until stabilizing them. We applied FC graph to some open source programs and visualized the results. In the author’s experience, FC graph is helpful to grasp correspondence of code clones over multiple files and also code clones with in a file.

Keywords: code clone detection, program comprehension, software maintenance, visualization

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Authors: Yilun Shang

Abstract:

In this paper, we investigate the appearance of the giant component in random subgraphs G(p) of a given large finite graph family Gn = (Vn, En) in which each edge is present independently with probability p. We show that if the graph Gn satisfies a weak isoperimetric inequality and has bounded degree, then the probability p under which G(p) has a giant component of linear order with some constant probability is bounded away from zero and one. In addition, we prove the probability of abnormally large order of the giant component decays exponentially. When a contact graph is modeled as Gn, our result is of special interest in the study of the spread of infectious diseases or the identification of community in various social networks.

Keywords: subgraph, expander, random graph, giant component, percolation.

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Authors: Erdoğan Özbay, Hakan T. Türker, Müzeyyen Balçıkanlı, Mohamed Lachemi

Abstract:

In this paper, the effects of fiber types and elevated temperatures on compressive strength, modulus of rapture and the bond characteristics of fiber reinforced concretes (FRC) are presented. By using the three different types of fibers (steel fiber-SF, polypropylene-PPF and polyvinyl alcohol-PVA), FRC specimens were produced and exposed to elevated temperatures up to 800 ºC for 1.5 hours. In addition, a plain concrete (without fiber) was produced and used as a control. Test results obtained showed that the steel fiber reinforced concrete (SFRC) had the highest compressive strength, modulus of rapture and bond stress values at room temperatures, the residual bond, flexural and compressive strengths of both FRC and plain concrete dropped sharply after exposure to high temperatures. The results also indicated that the reduction of bond, flexural and compressive strengths with increasing the exposed temperature was relatively less for SFRC than for plain, and FRC with PPF and PVA.

Keywords: Bond stress, Compressive strength, Elevated temperatures, Fiber reinforced concrete, Modulus of rapture.

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Authors: Adrien Derock, Pascal Veron

Abstract:

Taking into account the link between the efficiency of a detector and the complexity of a stealth mechanism, we propose in this paper a new formalism for stealth using graph theory.

Keywords: Detection, eradication, graph, rootkit, stealth.

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Authors: Brahim Hamid, Bertrand Le Saec, Mohamed Mosbah

Abstract:

The vertex connectivity of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. This work is devoted to the problem of vertex connectivity test of graphs in a distributed environment based on a general and a constructive approach. The contribution of this paper is threefold. First, using a preconstructed spanning tree of the considered graph, we present a protocol to test whether a given graph is 2-connected using only local knowledge. Second, we present an encoding of this protocol using graph relabeling systems. The last contribution is the implementation of this protocol in the message passing model. For a given graph G, where M is the number of its edges, N the number of its nodes and Δ is its degree, our algorithms need the following requirements: The first one uses O(Δ×N2) steps and O(Δ×logΔ) bits per node. The second one uses O(Δ×N2) messages, O(N2) time and O(Δ × logΔ) bits per node. Furthermore, the studied network is semi-anonymous: Only the root of the pre-constructed spanning tree needs to be identified.

Keywords: Distributed computing, fault-tolerance, graph relabeling systems, local computations, local knowledge, message passing system, networks, vertex connectivity.

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Authors: Imen Ammari, Mourad Elloumi, Ala Eddine Barouni

Abstract:

Visualizing “Courses – Pre – Required - Architecture" on the screen has proven to be useful and helpful for university actors and specially for students. In fact, these students can easily identify courses and their pre required, perceive the courses to follow in the future, and then can choose rapidly the appropriate course to register in. Given a set of courses and their prerequired, we present an algorithm for visualization a graph entitled “Courses-Pre-Required-Graph" that present courses and their prerequired in order to help students to recognize, lonely, what courses to take in the future and perceive the contain of all courses that they will study. Our algorithm using “Force Directed Placement" technique visualizes the “Courses-Pre-Required-Graph" in such way that courses are easily identifiable. The time complexity of our drawing algorithm is O (n2), where n is the number of courses in the “Courses-Pre-Required-Graph".

Keywords: Courses–Pre-Required-Architecture, Courses-Pre- Required-Graph, Courses-Pre-Required-Visualization, Force directed Placement, Resolution.

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Authors: Z.Farhadpour, Mohammad.R.Meybodi

Abstract:

A number of routing algorithms based on learning automata technique have been proposed for communication networks. How ever, there has been little work on the effects of variation of graph scarcity on the performance of these algorithms. In this paper, a comprehensive study is launched to investigate the performance of LASPA, the first learning automata based solution to the dynamic shortest path routing, across different graph structures with varying scarcities. The sensitivity of three main performance parameters of the algorithm, being average number of processed nodes, scanned edges and average time per update, to variation in graph scarcity is reported. Simulation results indicate that the LASPA algorithm can adapt well to the scarcity variation in graph structure and gives much better outputs than the existing dynamic and fixed algorithms in terms of performance criteria.

Keywords: Learning automata, routing, algorithm, sparse graph

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Authors: Maruful H. Mazumder, Raymond I. Gilbert

Abstract:

The design specifications for calculating development and lapped splice lengths of reinforcement in concrete are derived from a conventional empirical modelling approach that correlates experimental test data using a single mathematical equation. This paper describes part of a recently completed experimental research program to assess the effects of different structural parameters on the development length requirements of modern high strength steel reinforcing bars, including the case of lapped splices in large-scale reinforced concrete members. The normalized average bond stresses for the different variations of anchorage lengths are assessed according to the general form of a typical empirical analytical model of bond and anchorage. Improved analytical modelling equations are developed in the paper that better correlate the normalized bond strength parameters with the structural parameters of an empirical model of bond and anchorage.

Keywords: Bond stress, Development length, Lapped splice length, Reinforced concrete.

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Authors: Yingqi Cui, Changran Huang, Raymond Lee

Abstract:

In this paper, we proposed an AI Tutor using ontology and natural language process techniques to generate a computer science domain knowledge graph and answer users’ questions based on the knowledge graph. We define eight types of relation to extract relationships between entities according to the computer science domain text. The AI tutor is separated into two agents: learning agent and Question-Answer (QA) agent and developed on JADE (a multi-agent system) platform. The learning agent is responsible for reading text to extract information and generate a corresponding knowledge graph by defined patterns. The QA agent can understand the users’ questions and answer humans’ questions based on the knowledge graph generated by the learning agent.

Keywords: Artificial intelligence, natural language process, knowledge graph, agent, QA system.

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Authors: Jeffrey L. Duffany

Abstract:

Graph coloring is an important problem in computer science and many algorithms are known for obtaining reasonably good solutions in polynomial time. One method of comparing different algorithms is to test them on a set of standard graphs where the optimal solution is already known. This investigation analyzes a set of 50 well known graph coloring instances according to a set of complexity measures. These instances come from a variety of sources some representing actual applications of graph coloring (register allocation) and others (mycieleski and leighton graphs) that are theoretically designed to be difficult to solve. The size of the graphs ranged from ranged from a low of 11 variables to a high of 864 variables. The method used to solve the coloring problem was the square of the adjacency (i.e., correlation) matrix. The results show that the most difficult graphs to solve were the leighton and the queen graphs. Complexity measures such as density, mobility, deviation from uniform color class size and number of block diagonal zeros are calculated for each graph. The results showed that the most difficult problems have low mobility (in the range of .2-.5) and relatively little deviation from uniform color class size.

Keywords: graph coloring, complexity, algorithm.

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Authors: Mustafa Elfituri, Jonathan Cook

Abstract:

Recently, graph-based computations have become more important in large-scale scientific computing as they can provide a methodology to model many types of relations between independent objects. They are being actively used in fields as varied as biology, social networks, cybersecurity, and computer networks. At the same time, graph problems have some properties such as irregularity and poor locality that make their performance different than regular applications performance. Therefore, parallelizing graph algorithms is a hard and challenging task. Initial evidence is that standard computer architectures do not perform very well on graph algorithms. Little is known exactly what causes this. The Graph500 benchmark is a representative application for parallel graph-based computations, which have highly irregular data access and are driven more by traversing connected data than by computation. In this paper, we present results from analyzing the performance of various example implementations of Graph500, including a shared memory (OpenMP) version, a distributed (MPI) version, and a hybrid version. We measured and analyzed all the factors that affect its performance in order to identify possible changes that would improve its performance. Results are discussed in relation to what factors contribute to performance degradation.

Keywords: Graph computation, Graph500 benchmark, parallel architectures, parallel programming, workload characterization.

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Authors: A. Al-Mosawe, R. Al-Mahaidi

Abstract:

Carbon fiber reinforced polymersarewidely used to strengthen steel structural elements. These structural elements are normally subjected to static, dynamic and fatigue loadings during their life-time. CFRP laminate is commonly used to strengthen these structures under the subjected loads. A number of studies have focused on the characteristics of CFRP sheets bonded to steel members under static, dynamic and fatigue loadings. However, there is a gap in understanding the bonding behavior between CFRP laminates and steel members under impact loading. This paper shows the effect of high load rates on this bond. CFRP laminate CFK 150/2000 was used to strengthen steel joints using Araldite 420 epoxy. The results show that applying a high load rate significantly affects the bond strength but has little influence on the effective bond length.

Keywords: Adhesively-bonded joints, Bond strength, CFRP laminate, Impact tensile loading.

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Authors: S. Balaji, V. Swaminathan, K. Kannan

Abstract:

The Maximum Weighted Independent Set (MWIS) problem is a classic graph optimization NP-hard problem. Given an undirected graph G = (V, E) and weighting function defined on the vertex set, the MWIS problem is to find a vertex set S V whose total weight is maximum subject to no two vertices in S are adjacent. This paper presents a novel approach to approximate the MWIS of a graph using minimum weighted vertex cover of the graph. Computational experiments are designed and conducted to study the performance of our proposed algorithm. Extensive simulation results show that the proposed algorithm can yield better solutions than other existing algorithms found in the literature for solving the MWIS.

Keywords: weighted independent set, vertex cover, vertex support, heuristic, NP - hard problem.

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Authors: Pang Ying Han, Hiew Fu San, Ooi Shih Yin

Abstract:

Linearization of graph embedding has been emerged as an effective dimensionality reduction technique in pattern recognition. However, it may not be optimal for nonlinearly distributed real world data, such as face, due to its linear nature. So, a kernelization of graph embedding is proposed as a dimensionality reduction technique in face recognition. In order to further boost the recognition capability of the proposed technique, the Fisher-s criterion is opted in the objective function for better data discrimination. The proposed technique is able to characterize the underlying intra-class structure as well as the inter-class separability. Experimental results on FRGC database validate the effectiveness of the proposed technique as a feature descriptor.

Keywords: Face recognition, Fisher discriminant, graph embedding, kernelization.

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Authors: R. Anitha, R. S. Lekshmi

Abstract:

Graph decompositions are vital in the study of combinatorial design theory. Given two graphs G and H, an H-decomposition of G is a partition of the edge set of G into disjoint isomorphic copies of H. An n-sun is a cycle Cn with an edge terminating in a vertex of degree one attached to each vertex. In this paper we have proved that the complete graph of order 2n, K2n can be decomposed into n-2 n-suns, a Hamilton cycle and a perfect matching, when n is even and for odd case, the decomposition is n-1 n-suns and a perfect matching. For an odd order complete graph K2n+1, delete the star subgraph K1, 2n and the resultant graph K2n is decomposed as in the case of even order. The method of building n-suns uses Walecki's construction for the Hamilton decomposition of complete graphs. A spanning tree decomposition of even order complete graphs is also discussed using the labeling scheme of n-sun decomposition. A complete bipartite graph Kn, n can be decomposed into n/2 n-suns when n/2 is even. When n/2 is odd, Kn, n can be decomposed into (n-2)/2 n-suns and a Hamilton cycle.

Keywords: Hamilton cycle, n-sun decomposition, perfectmatching, spanning tree.

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Authors: Elsayed A. Habib Elamir

Abstract:

In social network analysis the mean nodal degree and density of the graph can be considered as a measure of the activity of all actors in the network and this is an important property of a graph and for making comparisons among networks. Since subjects in a family or organization are subject to common environment factors, it is prime interest to study the association between responses. Therefore, we study the distribution of the mean nodal degree and density of the graph under correlated binary units. The cross product ratio is used to capture the intra-units association among subjects. Computer program and an application are given to show the benefits of the method.

Keywords: Correlated Binary data, cross product ratio, densityof the graph, multiplicative binomial distribution.

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Authors: Shin-Shin Kao, Hsiu-Chunj Pan

Abstract:

Given a graph G. A cycle of G is a sequence of vertices of G such that the first and the last vertices are the same. A hamiltonian cycle of G is a cycle containing all vertices of G. The graph G is k-ordered (resp. k-ordered hamiltonian) if for any sequence of k distinct vertices of G, there exists a cycle (resp. hamiltonian cycle) in G containing these k vertices in the specified order. Obviously, any cycle in a graph is 1-ordered, 2-ordered and 3- ordered. Thus the study of any graph being k-ordered (resp. k-ordered hamiltonian) always starts with k = 4. Most studies about this topic work on graphs with no real applications. To our knowledge, the chordal ring families were the first one utilized as the underlying topology in interconnection networks and shown to be 4-ordered. Furthermore, based on our computer experimental results, it was conjectured that some of them are 4-ordered hamiltonian. In this paper, we intend to give some possible directions in proving the conjecture.

Keywords: Hamiltonian cycle, 4-ordered, Chordal rings, 3-regular.

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Authors: P.W. C. Prasad, A. Assi, A. Harb, V.C. Prasad

Abstract:

This paper presents an improved variable ordering method to obtain the minimum number of nodes in Reduced Ordered Binary Decision Diagrams (ROBDD). The proposed method uses the graph topology to find the best variable ordering. Therefore the input Boolean function is converted to a unidirectional graph. Three levels of graph parameters are used to increase the probability of having a good variable ordering. The initial level uses the total number of nodes (NN) in all the paths, the total number of paths (NP) and the maximum number of nodes among all paths (MNNAP). The second and third levels use two extra parameters: The shortest path among two variables (SP) and the sum of shortest path from one variable to all the other variables (SSP). A permutation of the graph parameters is performed at each level for each variable order and the number of nodes is recorded. Experimental results are promising; the proposed method is found to be more effective in finding the variable ordering for the majority of benchmark circuits.

Keywords: Binary decision diagrams, graph representation, Boolean functions representation, variable ordering.

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Authors: Wongsakorn Charoenpanitseri

Abstract:

The total chromatic number χ"(G) of a graph G is the minimum number of colors needed to color the elements (vertices and edges) of G such that no incident or adjacent pair of elements receive the same color Let G be a graph with maximum degree Δ(G). Considering a total coloring of G and focusing on a vertex with maximum degree. A vertex with maximum degree needs a color and all Δ(G) edges incident to this vertex need more Δ(G) + 1 distinct colors. To color all vertices and all edges of G, it requires at least Δ(G) + 1 colors. That is, χ"(G) is at least Δ(G) + 1. However, no one can find a graph G with the total chromatic number which is greater than Δ(G) + 2. The Total Coloring Conjecture states that for every graph G, χ"(G) is at most Δ(G) + 2. In this paper, we prove that the Total Coloring Conjectur for a Δ-claw-free 3-degenerated graph. That is, we prove that the total chromatic number of every Δ-claw-free 3-degenerated graph is at most Δ(G) + 2.

Keywords: Total colorings, the total chromatic number, 3-degenerated.

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