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15
Variable neighborhood search: Principles and applications
, 2001
"... Systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization, called variable neighborhood search (VNS). We present a basic scheme for this purpose, which can easily be implemented using an ..."
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Cited by 180 (17 self)
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Systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization, called variable neighborhood search (VNS). We present a basic scheme for this purpose, which can easily be implemented using any local search algorithm as a subroutine. Its effectiveness is illustrated by solving several classical combinatorial or global optimization problems. Moreover, several extensions are proposed for solving large problem instances: using VNS within the successive approximation method yields a twolevel VNS, called variable neighborhood decomposition search (VNDS); modifying the basic scheme to explore easily valleys far from the incumbent solution yields an efficient skewed VNS (SVNS) heuristic. Finally, we show how to stabilize column generation algorithms with help of VNS and discuss various ways to use VNS in graph theory, i.e., to suggest, disprove or give hints on how to prove conjectures, an area where metaheuristics do not appear
Bridges between Geometry and Graph Theory
 in Geometry at Work, C.A. Gorini, ed., MAA Notes 53
"... Graph theory owes many powerful ideas and constructions to geometry. Several wellknown families of graphs arise as intersection graphs of certain geometric objects. Skeleta of polyhedra are natural sources of graphs. Operations on polyhedra and maps give rise to various interesting graphs. Another ..."
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Cited by 15 (8 self)
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Graph theory owes many powerful ideas and constructions to geometry. Several wellknown families of graphs arise as intersection graphs of certain geometric objects. Skeleta of polyhedra are natural sources of graphs. Operations on polyhedra and maps give rise to various interesting graphs. Another source of graphs are geometric configurations where the relation of incidence determines the adjacency in the graph. Interesting graphs possess some inner structure which allows them to be described by labeling smaller graphs. The notion of covering graphs is explored.
Generation and properties of snarks
 Journal of Combinatorial Theory, Series B
"... Abstract. For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for snarks, the class of nontrivial 3regular graphs which cannot be 3edge coloured. In the first part of this paper we present a new algorithm for generating all n ..."
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Cited by 8 (3 self)
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Abstract. For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for snarks, the class of nontrivial 3regular graphs which cannot be 3edge coloured. In the first part of this paper we present a new algorithm for generating all nonisomorphic snarks of a given order. Our implementation of the new algorithm is 14 times faster than previous programs for generating snarks, and 29 times faster for generating weak snarks. Using this program we have generated all nonisomorphic snarks on n ≤ 36 vertices. Previously lists up to n = 28 vertices have been published. In the second part of the paper we analyze the sets of generated snarks with respect to a number of properties and conjectures. We find that some of the strongest versions of the cycle double cover conjecture hold for all snarks of these orders, as does Jaeger’s Petersen colouring conjecture, which in turn implies that Fulkerson’s conjecture has no
ON STABLE CYCLES AND CYCLE DOUBLE COVERS OF GRAPHS WITH LARGE CIRCUMFERENCE
"... Abstract. A cycle C in a graph is called stable if there exist no other cycle D in the same graph such that V (C) ⊆ V (D). In this paper we study stable cycles in snarks and based on our findings we are able to show that if a cubic graph G has a cycle of length at least V (G)  − 9 then it has a ..."
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Cited by 4 (2 self)
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Abstract. A cycle C in a graph is called stable if there exist no other cycle D in the same graph such that V (C) ⊆ V (D). In this paper we study stable cycles in snarks and based on our findings we are able to show that if a cubic graph G has a cycle of length at least V (G)  − 9 then it has a cycle double cover. We also give a construction for an infinite snark family with stable cycles of constant length and answer a question by Kochol by giving examples of cyclically 5edge connected cubic graphs with stable cycles. 1.
Computers and Discovery in Algebraic Graph Theory
 Edinburgh, 2001), Linear Algebra Appl
, 2001
"... We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory. ..."
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Cited by 4 (0 self)
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We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.
Graph structural properties of nonyutsis graphs allowing fast recognition
 Discrete Appl. Math
"... Yutsis graphs are connected simple graphs which can be partitioned into two vertexinduced trees. Cubic Yutsis graphs were introduced by Jaeger as cubic dual Hamiltonian graphs, and these are our main focus. Cubic Yutsis graphs also appear in the context of the quantum theory of angular momenta, whe ..."
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Cited by 2 (1 self)
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Yutsis graphs are connected simple graphs which can be partitioned into two vertexinduced trees. Cubic Yutsis graphs were introduced by Jaeger as cubic dual Hamiltonian graphs, and these are our main focus. Cubic Yutsis graphs also appear in the context of the quantum theory of angular momenta, where they are used to generate summation formulae for general recoupling coefficients. Large Yutsis graphs are of interest for benchmarking algorithms which generate these formulae. In an earlier paper we showed that the decision problem of whether a given cubic graph is Yutsis is NPcomplete. We also described a heuristic that was tested on graphs with up to 300,000 vertices and found Yutsis decompositions for all large Yutsis graphs very quickly. In contrast, no fast technique was known by which a significant fraction of bridgeless nonYutsis cubic graphs could be shown to be nonYutsis. One of the contributions of this article is to describe some structural impediments to Yutsisness and to provide experimental evidence that almost all nonYutsis cubic graphs can be rapidly shown to be nonYutsis by their application. Combined with the algorithm Preprint submitted to Discrete Applied Mathematics 5 October 2007 described in the earlier paper this gives an algorithm that, according to experimental evidence, runs efficiently on practically every large random cubic graph and can decide on whether the graph is Yutsis or not. The second contribution of this article is a set of construction techniques for nonYutsis graphs implying, for example, the existence of 3connected nonYutsis cubic graphs of arbitrary girth and with few nontrivial 3cuts. Key words: Yutsis graph; dual Hamiltonian graph; decision problem; general recoupling coefficient
Traceminimal graphs and Doptimal weighing designs, preprint
"... Let G(v, δ) be the set of all δregular graphs on v vertices. Certain graphs from among those in G(v, δ) with maximum girth have a special property called traceminimality. In particular, all strongly regular graphs with no triangles and some cages are traceminimal. These graphs play an important r ..."
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Let G(v, δ) be the set of all δregular graphs on v vertices. Certain graphs from among those in G(v, δ) with maximum girth have a special property called traceminimality. In particular, all strongly regular graphs with no triangles and some cages are traceminimal. These graphs play an important role in the statistical theory of Doptimal weighing designs. Each weighing design can be associated with a (0, 1)matrix. Let Mm,n(0, 1) denote the set of all m × n (0,1)matrices and let G(m, n) = max{det X T X: X ∈ Mm,n(0, 1)}. A matrix X ∈ Mm,n(0, 1) is a Doptimal design matrix if det X T X = G(m, n). In this paper we exhibit some new formulas for G(m, n) where n ≡ −1 (mod 4) and m is sufficiently large. These formulas depend on the congruence class of m (mod n). More precisely, let m = nt + r where 0 ≤ r < n. For each pair n, r, there is a polynomial P (n, r, t) of degree n in t, which depends only on n, r, such that G(nt + r, n) = P (n, r, t) for all sufficiently large t. The polynomial P (n, r, t) is computed from the characteristic polynomial of the adjacency matrix of a traceregular graph whose degree of regularity and number of vertices depend only on n and r. We obtain explicit expressions for the polynomial P (n, r, t) for many pairs n, r. In particular we obtain formulas for G(nt + r, n) for n = 19, 23, and 27, all 0 ≤ r < n, and all sufficiently large t. And we obtain families of formulas for P (n, r, t) from families of traceminimal graphs including bipartite graphs obtained from finite projective planes, generalized quadrilaterals, and generalized hexagons. Keywords: Doptimal weighing design, traceminimal graph, regular graph, strongly regular graph, girth, cages, generalized polygons AMS Subject Classification:
The Complete Catalog of 3Regular, Diameter3 Planar Graphs
, 1996
"... The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 ..."
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Cited by 1 (1 self)
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The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 vertices. An Upper Bound A graph with maximum degree \Delta and diameter D is called a (\Delta; D)graph. It is easily seen ([9], p. 171) that the order of a (\Delta,D)graph is bounded above by the Moore bound, which is given by 1+ \Delta + \Delta (\Delta \Gamma 1) + \Delta \Delta \Delta + \Delta(\Delta \Gamma 1) D\Gamma1 = 8 ? ! ? : \Delta(\Delta \Gamma 1) D \Gamma 2 \Delta \Gamma 2 if \Delta 6= 2; 2D + 1 if \Delta = 2: Figure 1: The regular (3,3)graph on 20 vertices (it is unique up to isomorphism) . For D 2 and \Delta 3, this bound is attained only if D = 2 and \Delta = 3; 7, and (perhaps) 57 [3, 14, 23]. Now, except for the case of C 4 (the cycle on four vertices), the num...
Cyclic Haar Graphs
, 1999
"... For a given group \Gamma with a generating set A, a dipole with jAj parallel ..."
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Cited by 1 (0 self)
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For a given group \Gamma with a generating set A, a dipole with jAj parallel