5666902
The chromatic numbers of distance graphs
Abstract Suppose D is a subset of all positive integers L . The distance graph G ( L , D ) is the graph with vertex set L and two vertices x , y are adjacent if | x − y | ∈ D . This paper studies the chromatic number of G ( L , D ). We shall concentrate on the case that | D | = 3 and give a complete classification of those D = { a , b , c } for which b is a multiple of a and χ ( G ( L , D )) = 3.
243486
Circular chromatic index of graphs of maximum degree 3
This paper proves that if $G$ is a graph (parallel edges allowed) of maximum degree 3, then $\chi_c'(G) \leq 11/3$ provided that $G$ does not contain $H_1$ or $H_2$ as a subgraph, where $H_1$ and $H_2$ are obtained by subdividing one edge of $K_2^3$ (the graph with three parallel edges between two vertices) and $K_4$, respectively. As $\chi_c'(H_1) = \chi_c'(H_2) = 4$, our result implies that there is no graph $G$ with $11/3 < \chi_c'(G) < 4$. It also implies that if $G$ is a 2-edge connected cubic graph, then $\chi'(G) \le 11/3$.
7791840
Pseudo-Hamiltonian-connected graphs
Abstract Given a graph G and a positive integer k , denote by G[k] the graph obtained from G by replacing each vertex of G with an independent set of size k . A graph G is called pseudo- k Hamiltonian-connected if G[k] is Hamiltonian-connected, i.e., every two distinct vertices of G[k] are connected by a Hamiltonian path. A graph G is called pseudo Hamiltonian-connected if it is pseudo- k Hamiltonian-connected for some positive integer k . This paper proves that a graph G is pseudo-Hamiltonian-connected if and only if for every non-empty proper subset X of V(G) , |N(X)|>|X| . The proof of the characterization also provides a polynomial-time algorithm that decides whether or not a given graph is pseudo-Hamiltonian-connected. The characterization of pseudo-Hamiltonian-connected graphs also answers a question of Richard Nowakowski, which motivated this paper.
5956515
Homomorphisms to oriented paths
Abstract Given an oriented path P , we characterize those digraphs G which are homomorphic to P . The characterization equates the nonexistence of a homomorphism G → P with the existence of a homomorphism W → G , for some oriented path W which is not homomorphic to P . This result complements the recent polynomial time algorithm of Welzl et al. to find such a homomorphism (if one exists).
2178223
Sparse H-Colourable Graphs of Bounded Maximum Degree
Abstract.Let F be a graph of order at most k. We prove that for any integer g there is a graph G of girth at least g and of maximum degree at most 5k13 such that G admits a surjective homomorphism c to F, and moreover, for any F-pointed graph H with at most k vertices, and for any homomorphism h from G to H there is a unique homomorphism f from F to H such that h=f∘c. As a consequence, we prove that if H is a projective graph of order k, then for any finite family of prescribed mappings from a set X to V(H) (with ||=t), there is a graph G of arbitrary large girth and of maximum degree at most 5k26mt (where m=|X|) such that and up to an automorphism of H, there are exactly t homomorphisms from G to H, each of which is an extension of an f.
206792834
Star chromatic numbers of graphs
We investigate the relation between the star-chromatic number χ(G) and the chromatic number χ(G) of a graphG. First we give a sufficient condition for graphs under which their starchromatic numbers are equal to their ordinary chromatic numbers. As a corollary we show that for any two positive integersk, g, there exists ak-chromatic graph of girth at leastg whose star-chromatic number is alsok. The special case of this corollary withg=4 answers a question of Abbott and Zhou. We also present an infinite family of triangle-free planar graphs whose star-chromatic number equals their chromatic number. We then study the star-chromatic number of An infinite family of graphs is constructed to show that for each ε>0 and eachm≥2 there is anm-connected (m+1)-critical graph with star chromatic number at mostm+ε. This answers another question asked by Abbott and Zhou.
10041607
Comparison of Modularization Methods in Application to Different Biological Networks
Most biological networks have been proposed to possess modular organization, which increases the robustness, flexibility, and stability of networks. Many clustering methods have been used in mining biological data and partitioning complex networks into functional modules. Most of these methods require presetting the number of modules and therefore can potentially obtain biased results. The Markov clustering method (MCL) and the simulated annealing module-detection method (SA) eliminate this requirement and can objectively separate relatively dense subgraphs. In this paper, we compared these two module-detection methods for three types of biological data: protein family classification, microarray clustering, and modularity of metabolic networks. We found that these two methods show differential advantages for different biological networks. In the case of the gene network based on Affymetrix microarray spike data, MCL exactly identified the same number of groups and same contents in each group set by the spike data. In the case of the gene network derived from actual expression data, although neither of the two methods can perfectly recover the natural classification, MCL performs slightly better than SA. However, with increased random noise added to the gene expression values, SA generates better modular structures with higher modularity. Next we compared the modularization results of MCL and SA for protein family classification and found the modules detected by SA could not be well matched with the Structural Classification of Proteins (SCOP database), which suggests that MCL is ideally suited to the rapid and accurate detection of protein families. In addition, we used both methods to detect modules in the metabolic network of E. coli. MCL gives a trivial clustering, which generates biologically insignificant modules. In contrast, SA detects modules well corresponding to the KEGG functional classification. Moreover the modularity for several other metabolic networks detected by SA is also much higher than that by MCL. In summary, MCL is more suited to modularize relatively complete and definite data, such as a protein family network. In contrast, SA is less sensitive to noise such as experimental error or incomplete data and outperforms MCL when modularizing gene networks based on microarray data and large scale metabolic networks constructed from incomplete databases.
205872736
Flatness tolerance evaluation: an approximate minimum zone solution
Abstract This paper is concerned with the problem of flatness tolerance evaluation. First, we introduce the concepts of width and inner radius of point sets, and establish the equivalence between the width of a point set and the inner radius of the convex hull of the self-difference of the set. On this basis, we present an algorithm for calculating the exact minimum zone tolerance of flatness. Second, we prove that an approximation of the inner radius of a convex set can be obtained from the inner radius of the set in fixed direction. On this basis, we present an algorithm for calculating the ‘almost exact’ minimum zone solution. This algorithm is implemented by solving a single linear programming problem, of which the computational complexity is O ( n ). Numerical examples are given to validate the proposed algorithms.
5944615
Eliminating Class Noise in Large Datasets
This paper presents a new approach for identifying and eliminating mislabeled instances in large or distributed datasets. We first partition a dataset into subsets, each of which is small enough to be processed by an induction algorithm at one time. We construct good rules from each subset, and use the good rules to evaluate the whole dataset. For a given instance Ik, two error count variables are used to count the number of times it has been identified as noise by all subsets. The instance with higher error values will have a higher probability of being a mislabeled example. Two threshold schemes, majority and non-objection, are used to identify the noise. Experimental results and comparative studies from real-world datasets are reported to evaluate the effectiveness and efficiency of the proposed approach.
739919
Orion: a power-performance simulator for interconnection networks
With the prevalence of server blades and systems-on-a-chip (SoCs), interconnection networks are becoming an important part of the microprocessor landscape. However, there is limited tool support available for their design. While performance simulators have been built that enable performance estimation while varying network parameters, these cover only one metric of interest in modern designs. System power consumption is increasingly becoming equally, if not more important than performance. It is now critical to get detailed power-performance tradeoff information early in the microarchitectural design cycle. This is especially so as interconnection networks consume a significant fraction of total system power. It is exactly this gap that the work presented in this paper aims to fill.We present Orion, a power-performance interconnection network simulator that is capable of providing detailed power characteristics, in addition to performance characteristics, to enable rapid power-performance trade-offs at the architectural-level. This capability is provided within a general framework that builds a simulator starting from a microarchitectural specification of the interconnection network. A key component of this construction is the architectural-level parameterized power models that we have derived as part of this effort. Using component power models and a synthesized efficient power (and performance) simulator, a microarchitect can rapidly explore the design space. As case studies, we demonstrate the use of Orion in determining optimal system parameters, in examining the effect of diverse traffic conditions, as well as evaluating new network microarchitectures. In each of the above, the ability to simultaneously monitor power and performance is key in determining suitable microarchitectures.
9200586
Scalable Inductive Learning on Partitioned Data
With the rapid advancement of information technology, scalability has become a necessity for learning algorithms to deal with large, real-world data repositories. In this paper, scalability is accomplished through a data reduction technique, which partitions a large data set into subsets, applies a learning algorithm on each subset sequentially or concurrently, and then integrates the learned results. Five strategies to achieve scalability (Rule-Example Conversion, Rule Weighting, Iteration, Good Rule Selection, and Data Dependent Rule Selection) are identified and seven corresponding scalable schemes are designed and developed. A substantial number of experiments have been performed to evaluate these schemes. Experimental results demonstrate that through data reduction some of our schemes can effectively generate accurate classifiers from weak classifiers generated from data subsets. Furthermore, our schemes require significantly less training time than that of generating a global classifier.