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37
Partitioning into graphs with only small components
 J. Combin. Theory Ser. B
, 2003
"... Abstract. The paper presents several results on edge partitions and vertex partitions of graphs into graphs with bounded size components. We show that every graph of bounded treewidth and bounded maximum degree admits such partitions. We also show that an arbitrary graph of maximum degree four has ..."
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Cited by 20 (0 self)
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Abstract. The paper presents several results on edge partitions and vertex partitions of graphs into graphs with bounded size components. We show that every graph of bounded treewidth and bounded maximum degree admits such partitions. We also show that an arbitrary graph of maximum degree four has a vertex partition into two graphs, each of which has components on at most 57 vertices. Some generalizations of the last result are also discussed. 1.
From a zoo to a zoology: Towards a general theory of graph polynomials
 Theory of Computing Systems
, 2007
"... Abstract. We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We in ..."
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Cited by 13 (4 self)
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Abstract. We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce two classes of (hyper)graph polynomials definable in second order logic, and outline a research program for their classification in terms of definability and complexity considerations, and various notions of reducibilities. 1
An optimized reconfigurable architecture for Transputer networks
 Proc. of 25th Hawaii Int. Conf. on System Sciences (HICSS 92
, 1992
"... This paper presents the architecture of a fully reconfigurable distributed memory computing system. It is assumed that the processors communicate via message passing on an application specific regular network of degree four. To realize any network of this class, we use a special multistage Clos netw ..."
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Cited by 10 (3 self)
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This paper presents the architecture of a fully reconfigurable distributed memory computing system. It is assumed that the processors communicate via message passing on an application specific regular network of degree four. To realize any network of this class, we use a special multistage Clos network which is built up by a minimal number of equal sized switches. These switches can be configured to realize any connection between input and output ports. To map a network onto the architecture, the process graph has to be partitioned into a number of subsets. We prove that the number of external edges between the subsets can be bounded. For that reason, it is possible to minimize the number of links and switches in our architecture without loosing the ability to realize any regular network of degree four. Moreover, any user specific network can be mapped efficiently on the architecture. This implies an efficient configuration of the system. The multistage structure of the architecture ma...
A Revival of the Girth Conjecture
, 2003
"... We show that for each " > 0, there exists a number g such that the circular chromatic index of every cubic bridgeless graph with girth at least g is at most 3 + ". This contrasts to the fact (which disproved the Girth Conjecture) that there are snarks of arbitrary large girth. In particular, ..."
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Cited by 8 (3 self)
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We show that for each " > 0, there exists a number g such that the circular chromatic index of every cubic bridgeless graph with girth at least g is at most 3 + ". This contrasts to the fact (which disproved the Girth Conjecture) that there are snarks of arbitrary large girth. In particular,
Simultaneous diagonal flips in plane triangulations
 In Proc. 17th Annual ACMSIAM Symp. on Discrete Algorithms (SODA ’06
, 2006
"... Simultaneous diagonal flips in plane triangulations are investigated. It is proved that every nvertex triangulation with at least six vertices has a simultaneous flip into a 4connected triangulation, and that it can be computed in O(n) time. It follows that every triangulation has a simultaneous f ..."
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Cited by 7 (3 self)
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Simultaneous diagonal flips in plane triangulations are investigated. It is proved that every nvertex triangulation with at least six vertices has a simultaneous flip into a 4connected triangulation, and that it can be computed in O(n) time. It follows that every triangulation has a simultaneous flip into a Hamiltonian triangulation. This result is used to prove that for any two nvertex triangulations, there exists a sequence of O(log n) simultaneous flips to transform one into the other. The total number of edges flipped in this sequence is O(n). The maximum size of a simultaneous flip is then studied. It is proved that every triangulation has a simultaneous flip of at least 1 (n − 2) edges. On the other hand, every simultaneous flip has at most n − 2 edges, 3 and there exist triangulations with a maximum simultaneous flip of 6 (n − 2) edges. 7
On packing Hamilton Cycles in ɛRegular Graphs
, 2003
"... A graph G = (V; E) on n vertices is (; )regular if its minimal degree is at least n, and for every pair of disjoint subsets S; T V of cardinalities at least n, the number of edges e(S; T ) between S and T satis es: e(S;T ) . We prove that if > 0 are constants, then every (; )regular ..."
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Cited by 7 (6 self)
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A graph G = (V; E) on n vertices is (; )regular if its minimal degree is at least n, and for every pair of disjoint subsets S; T V of cardinalities at least n, the number of edges e(S; T ) between S and T satis es: e(S;T ) . We prove that if > 0 are constants, then every (; )regular graph on n vertices contains a family of (=2 O())n edgedisjoint Hamilton cycles. As a consequence we derive that for every constant 0 < p < 1, with high probability in the random graph G(n; p), almost all edges can be packed into edgedisjoint Hamilton cycles. A similar result is proven for the directed case.
Decidable Properties of Graphs of AllOptical Networks
"... We examine several decidability questions suggested by questions about alloptical networks, related to the gap between maximal load and number of colors (wavelengths) needed for a legal routing on a fixed graph. We prove the multiple fiber conjecture: for every fixed graph G there is a number LG ..."
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Cited by 6 (0 self)
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We examine several decidability questions suggested by questions about alloptical networks, related to the gap between maximal load and number of colors (wavelengths) needed for a legal routing on a fixed graph. We prove the multiple fiber conjecture: for every fixed graph G there is a number LG such that in the communication network with LG parallel fibers for each edge of G, there is no gap (for any load). We prove that for a fixed graph G the existence of a gap is computable, and give an algorithm to compute it. We develop a decomposition theory for paths, defining the notion of prime sets of paths that are finite building blocks for all loads on a fixed graph. Properties of such decompositions yield our theorems.
Partitioning Problems: Characterization, Complexity and Algorithms on Partial kTrees
, 1994
"... This thesis investigates the computational complexity of algorithmic problems defined on graphs. At the abstract level of the complexity spectrum we discriminate polynomialtime solvable problems from N Pcomplete problems, while at the concrete level we improve on polynomialtime algorithms for gen ..."
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Cited by 6 (1 self)
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This thesis investigates the computational complexity of algorithmic problems defined on graphs. At the abstract level of the complexity spectrum we discriminate polynomialtime solvable problems from N Pcomplete problems, while at the concrete level we improve on polynomialtime algorithms for generally hard problems restricted to treedecomposable graphs. One contribution of this thesis is a precise characterization of vertex partitioning problems which include variants of domination, coloring and packing. An elaboration of this characterization is given for problems defined over vertex subsets and over maximal/minimal vertex subsets. We introduce several new graph parameters as vertex partition generalizations of classical parameters. The given characterizations provide a basis for a taxonomy of vertex partitioning problems, facilitating their common algorithmic treatment and allowing for their uniform complexity classification. We explore the computational complexity of two important types of problems
Edgedisjoint Hamilton cycles in random graphs, preprint
"... Abstract. We show that provided log 50 n/n ≤ p ≤ 1 − n −1/4 log 9 n we can with high probability find a collection of ⌊δ(G)/2 ⌋ edgedisjoint Hamilton cycles in G ∼ Gn,p, plus an additional edgedisjoint matching of size ⌊n/2 ⌋ if δ(G) is odd. This is clearly optimal and confirms, for the above rang ..."
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Cited by 6 (3 self)
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Abstract. We show that provided log 50 n/n ≤ p ≤ 1 − n −1/4 log 9 n we can with high probability find a collection of ⌊δ(G)/2 ⌋ edgedisjoint Hamilton cycles in G ∼ Gn,p, plus an additional edgedisjoint matching of size ⌊n/2 ⌋ if δ(G) is odd. This is clearly optimal and confirms, for the above range of p, a conjecture of Frieze and Krivelevich. 1.
The circular chromatic index of graphs of high girth
 J. COMBIN. TH. (B
"... We show that for each ε>0 and each integer ∆ ≥ 1, there exists a number g such that for any graph G of maximum degree ∆ and girth at least g, the circular chromatic index of G is at most ∆ + ε. ..."
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Cited by 5 (2 self)
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We show that for each ε>0 and each integer ∆ ≥ 1, there exists a number g such that for any graph G of maximum degree ∆ and girth at least g, the circular chromatic index of G is at most ∆ + ε.