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Cayley Graphs and Interconnection Networks
, 1997
"... In this report, we will focus on routing problems including connectivity, diameter and loads of routings. We place the emphasis on load problems, since new classes of vertextransitive graphs (see quasiCayley graphs in Section 5.3) or edgetransitive graphs (see regular orbital graphs in Section 5. ..."
Abstract

Cited by 72 (3 self)
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In this report, we will focus on routing problems including connectivity, diameter and loads of routings. We place the emphasis on load problems, since new classes of vertextransitive graphs (see quasiCayley graphs in Section 5.3) or edgetransitive graphs (see regular orbital graphs in Section 5.6) were studied in order to enlarge the class of graphs behaving well for loads of routings. More precisely, after recalling some basic definitions and properties in Chapter 2, we survey particular classes of Cayley graphs which are often studied as models of interconnection networks. First, we study Cayley graphs generated by transpositions (Section 3.1) and those with a special automorphism which rotates the generators and is called a complete rotation (Section 3.2). In Section 3.3, we present hierarchical Cayley digraphs and results about fault tolerance. Finally, multistage Cayley digraphs and their quotients are studied in Section 3.4.
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 12 (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...
Clique Covering and Degree Conditions for Hamiltonicity in ClawFree Graphs
, 1997
"... By using the closure concept introduced by the last author, we prove that for any sufficiently large nonhamiltonian clawfree graph G satisfying a degree condition of type oe k (G) ? n + k 2 \Gamma 4k + 7 (where k is a constant), the closure of G can be covered by at most k \Gamma 1 cliques. Using ..."
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Cited by 8 (0 self)
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By using the closure concept introduced by the last author, we prove that for any sufficiently large nonhamiltonian clawfree graph G satisfying a degree condition of type oe k (G) ? n + k 2 \Gamma 4k + 7 (where k is a constant), the closure of G can be covered by at most k \Gamma 1 cliques. Using structural properties of the closure concept, we show a method for characterizing all such nonhamiltonian exceptional graphs with limited clique covering number. The method is explicitly carried out for k 6 and illustrated by proving that every 2connected clawfree graph G of order n 77 with ffi (G) 14 and oe 6 (G) ? n + 19 is either hamiltonian or belongs to a family of easily described exceptions.
*The workshop logo is based on a drawing of the HoffmanSingleton graph found by R.A. Litherland.
"... 14:45 15:25 Leif K. Jørgensen: Upper bounds for the degree/diameter problem 15:30 15:50 Minh Nguyen: On graphs of diameter two and order close to Moore bound 15:50 16:15 Coffee Break 16:15 16:45 Charles Delorme: Almost Moore bipartite graphs 16:50 17:10 Guillermo PinedaVillavicencio: New resul ..."
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14:45 15:25 Leif K. Jørgensen: Upper bounds for the degree/diameter problem 15:30 15:50 Minh Nguyen: On graphs of diameter two and order close to Moore bound 15:50 16:15 Coffee Break 16:15 16:45 Charles Delorme: Almost Moore bipartite graphs 16:50 17:10 Guillermo PinedaVillavicencio: New results on the degree/diameter problem
Factorcriticality and matching extension in DCTgraphs
 MATH. GRAPH THEORY
, 1996
"... The class of DCTgraphs is a common generalization of the classes of almost clawfree and quasi clawfree graphs. We prove that every even (2p + 1)connected DCTgraph G is pextendable, i.e. every set of p independent edges of G is contained in a perfect matching of G. This result is obtained as ..."
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The class of DCTgraphs is a common generalization of the classes of almost clawfree and quasi clawfree graphs. We prove that every even (2p + 1)connected DCTgraph G is pextendable, i.e. every set of p independent edges of G is contained in a perfect matching of G. This result is obtained as a corollary of a stronger result concerning factorcriticality of DCTgraphs.
Shortest Walks in Almost ClawFree Graphs
, 1993
"... There have been many results concerning clawfree graphs and hamiltonicity. Recently, Jackson and Wormald have obtained more general results on walks in clawfree graphs. In this paper, we consider the family of almost clawfree graphs that contains the previous one, and give some results on walk ..."
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Cited by 1 (1 self)
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There have been many results concerning clawfree graphs and hamiltonicity. Recently, Jackson and Wormald have obtained more general results on walks in clawfree graphs. In this paper, we consider the family of almost clawfree graphs that contains the previous one, and give some results on walks, especially on shortest covering walks visiting only once some given vertices.
Induced S(K 1,3) and Hamiltonian Square
, 1997
"... We prove that the square of a connected graph such that every induced S(K 1,3 ) has at least three edges in a block of degree at most 2 is hamiltonian. We also show that the insertion, and, under certain conditions also deletion, of a block of degree 2 into (from) a connected graph does not change t ..."
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We prove that the square of a connected graph such that every induced S(K 1,3 ) has at least three edges in a block of degree at most 2 is hamiltonian. We also show that the insertion, and, under certain conditions also deletion, of a block of degree 2 into (from) a connected graph does not change the hamiltonicity of its square.
Ore and Erdős type conditions for long cycles in balanced bipartite graphs
"... We conjecture Ore and Erdős type criteria for a balanced bipartite graph of order 2n to contain a long cycle C2n−2k, where 0 ≤ k < n/2. For k = 0, these are the classical hamiltonicity criteria of Moon and Moser. The main two results of the paper assert that our conjectures hold for k = 1 as wel ..."
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Cited by 1 (1 self)
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We conjecture Ore and Erdős type criteria for a balanced bipartite graph of order 2n to contain a long cycle C2n−2k, where 0 ≤ k < n/2. For k = 0, these are the classical hamiltonicity criteria of Moon and Moser. The main two results of the paper assert that our conjectures hold for k = 1 as well.
Results 1  10
of
32