Results 1  10
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21
A Survey of Combinatorial Gray Codes
 SIAM Review
, 1996
"... The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing nbit binary numbers so that ..."
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Cited by 99 (2 self)
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The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing nbit binary numbers so that successive numbers differ in exactly one bit position, as well as work in the 1960's and 70's on minimal change listings for other combinatorial families, including permutations and combinations. The area of combinatorial Gray codes was popularized by Herbert Wilf in his invited address at the SIAM Discrete Mathematics Conference in 1988 and his subsequent SIAM monograph in which he posed some open problems and variations on the theme. This resulted in much recent activity in the area and most of the problems posed by Wilf are now solved. In this paper, we survey the area of combinatorial Gray codes, describe recent results, variations, and trends, and highlight some open problems. ...
Generalized loopback recovery in optical mesh networks
 IEEE/ACM TRANSACTIONS ON NETWORKING
, 2002
"... Current means of providing loopback recovery, which is widely used in SONET, rely on ring topologies, or on overlaying logical ring topologies upon physical meshes. Loopback is desirable to provide rapid preplanned recovery of link or node failures in a bandwidthefficient distributed manner. We i ..."
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Cited by 32 (4 self)
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Current means of providing loopback recovery, which is widely used in SONET, rely on ring topologies, or on overlaying logical ring topologies upon physical meshes. Loopback is desirable to provide rapid preplanned recovery of link or node failures in a bandwidthefficient distributed manner. We introduce generalized loopback, a novel scheme for performing loopback in optical mesh networks. We present an algorithm to perform recovery for link failure and one to perform generalized loopback recovery for node failure. We illustrate the operation of both algorithms, prove their validity, and present a network management protocol algorithm, which enables distributed operation for link or node failure. We present three different applications of generalized loopback. First, we present heuristic algorithms for selecting recovery graphs, which maintain short maximum and average lengths of recovery paths. Second, we present WDMbased loopback recovery for optical networks where wavelengths are used to back up other wavelengths. We compare, for WDMbased loopback, the operation of generalized loopback operation with known ringbased ways of providing loopback recovery over mesh networks. Finally, we introduce the use of generalized loopback to provide recovery in a way that allows dynamic choice of routes over preplanned directions.
Sparse PseudoRandom Graphs Are Hamiltonian
 J. Graph Theory
, 2002
"... In this paper we study Hamilton cycles in sparse pseudorandom graphs. We prove that if the second largest absolute value λ of an eigenvalue of a dregular graph G on n vertices satisfies λ ≤ (log log n)² d / (1000 log n(log log log n)) and n is large enough, th ..."
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Cited by 29 (11 self)
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In this paper we study Hamilton cycles in sparse pseudorandom graphs. We prove that if the second largest absolute value &lambda; of an eigenvalue of a dregular graph G on n vertices satisfies &lambda; &le; (log log n)&sup2; d / (1000 log n(log log log n)) and n is large enough, then G is Hamiltonian.
Advances on the Hamiltonian problem  A survey
, 2002
"... This article is intended as a survey, updating earlier surveys in the area. For completeness of the presentation of both particular questions and the general area, it also contains material on closely related topics such as traceable, pancyclic and hamiltonianconnected graphs and digraphs. ..."
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Cited by 26 (0 self)
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This article is intended as a survey, updating earlier surveys in the area. For completeness of the presentation of both particular questions and the general area, it also contains material on closely related topics such as traceable, pancyclic and hamiltonianconnected graphs and digraphs.
Embedding large subgraphs into dense graphs
"... Abstract. What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac’s theorem on Hamilton cycles and Tutte’s theorem on perfect matchings. Perfect matchings are generalized by perfect Fpackings, where instead o ..."
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Cited by 19 (11 self)
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Abstract. What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac’s theorem on Hamilton cycles and Tutte’s theorem on perfect matchings. Perfect matchings are generalized by perfect Fpackings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect Fpacking, so as in the case of Dirac’s theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect Fpacking. The Regularity lemma of Szemerédi and the Blowup lemma of Komlós, Sárközy and Szemerédi have proved to be powerful tools in attacking such problems and quite recently, several longstanding problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on Fpackings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved.
The Antipodal Layers Problem
"... For n > 2k and [n] = f1; 2; : : : ; ng, let the bipartite graph M n;k have vertices fA [n] jAj = k or n kg and edges f(A; B) A Bg. It has been conjectured that M 2k+1;k (the middle two levels of the Boolean Lattice Q 2k+1 ) is Hamiltonian, and we conjecture the same for arbitrary n. ..."
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Cited by 12 (0 self)
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For n > 2k and [n] = f1; 2; : : : ; ng, let the bipartite graph M n;k have vertices fA [n] jAj = k or n kg and edges f(A; B) A Bg. It has been conjectured that M 2k+1;k (the middle two levels of the Boolean Lattice Q 2k+1 ) is Hamiltonian, and we conjecture the same for arbitrary n. Here we show that the conjecture holds for n bigger than roughly k 2 , with k large enough. We also dene a new product between ranked posets, giving rise to many new representations of M 2k+1;k . 1 1
ClawFree Graphs  a Survey.
, 1996
"... In this paper we summarize known results on clawfree graphs. The paper is subdivided into the following chapters and sections: 1. Introduction 2. Paths, cycles, hamiltonicity a) Preliminaries b) Degree and neighborhood conditions c) Local connectivity conditions d) Further forbidden subgraph ..."
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Cited by 11 (1 self)
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In this paper we summarize known results on clawfree graphs. The paper is subdivided into the following chapters and sections: 1. Introduction 2. Paths, cycles, hamiltonicity a) Preliminaries b) Degree and neighborhood conditions c) Local connectivity conditions d) Further forbidden subgraphs e) Invariants f) Squares g) Regular graphs h) Other hamiltonicity related results and generalizations 3. Matchings and factors 4. Independence, domination, other invariants and extremal problems 5. Algorithmic aspects 6. Miscellaneous 7. Appendix  List of all 2connected nonhamiltonian clawfree graphs on n 12 vertices. 1 This research was done while the first and second author were visiting Univ. of West Bohemia and while the third author was visiting L.R.I. and Memphis State University. 2 Research partially supported by ONR Grant N00014 91J1085 and NSA MDA 90490H4034. 3 Research supported by EC grant No.192493. 1 1. INTRODUCTION Clawfree graphs have been a su...
Controlled Generation of Hard and Easy Bayesian Networks: Impact on Maximal Clique Tree in Tree Clustering
 Artificial Intelligence
, 2006
"... This article presents and analyzes algorithms that systematically generate random Bayesian networks of varying difficulty levels, with respect to inference using tree clustering. The results are relevant to research on efficient Bayesian network inference, such as computing a most probable explanati ..."
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Cited by 9 (8 self)
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This article presents and analyzes algorithms that systematically generate random Bayesian networks of varying difficulty levels, with respect to inference using tree clustering. The results are relevant to research on efficient Bayesian network inference, such as computing a most probable explanation or belief updating, since they allow controlled experimentation to determine the impact of improvements to inference algorithms. The results are also relevant to research on machine learning of Bayesian networks, since they support controlled generation of a large number of data sets at a given difficulty level. Our generation algorithms, called BPART and MPART, support controlled but random construction of bipartite and multipartite Bayesian networks. The Bayesian network parameters that we vary are the total number of nodes, degree of connectivity, the ratio of the number of nonroot nodes to the number of root nodes, regularity of the underlying graph, and characteristics of the conditional probability tables. The main dependent parameter is the size of the maximal clique as generated by tree clustering. This article presents extensive empirical analysis using the H��� � tree clustering approach as well as theoretical analysis related to the random generation of Bayesian networks using BPART and MPART. 1
Weakly Pancyclic Graphs
, 1996
"... In generalizing the concept of a pancyclic graph, we say that a graph is `weakly pancyclic' if it contains cycles of every length between the length of a shortest and a longest cycle. In this paper it is shown that in many cases the requirements on a graph which ensure that it is weakly pancycl ..."
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Cited by 7 (0 self)
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In generalizing the concept of a pancyclic graph, we say that a graph is `weakly pancyclic' if it contains cycles of every length between the length of a shortest and a longest cycle. In this paper it is shown that in many cases the requirements on a graph which ensure that it is weakly pancyclic are considerably weaker than those required to ensure that it is pancyclic. This sheds some light on the content of a famous metaconjecture of Bondy. From the main result of this paper it follows that 2connected nonbipartite graphs of sufficiently large order n with minimum degree exceeding 2n/7 are weakly pancyclic; and that graphs with minimum degree at least n/4 + 250 are pancyclic, if they contain both a triangle and a hamiltonian cycle.
A SURVEY ON HAMILTON CYCLES IN DIRECTED GRAPHS
"... Abstract. We survey some recent results on longstanding conjectures regarding Hamilton cycles in directed graphs, oriented graphs and tournaments. We also combine some of these to prove the following approximate result towards Kelly’s conjecture on Hamilton decompositions of regular tournaments: th ..."
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Cited by 6 (6 self)
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Abstract. We survey some recent results on longstanding conjectures regarding Hamilton cycles in directed graphs, oriented graphs and tournaments. We also combine some of these to prove the following approximate result towards Kelly’s conjecture on Hamilton decompositions of regular tournaments: the edges of every regular tournament can be covered by a set of Hamilton cycles which are ‘almost ’ edgedisjoint. We also highlight the role that the notion of ‘robust expansion ’ plays in several of the proofs. New and old open problems are discussed. 1.