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A Constructive Enumeration of Fullerenes
"... In this paper, a fast and complete method to enumerate fullerene structures is given. It is based on a topdown approach, and it is fast enough to generate, for example, all 1812 isomers of C 60 in less than 20 seconds on an SGIworkstation. The method described can easily be generalised for 3regul ..."
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Cited by 14 (3 self)
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In this paper, a fast and complete method to enumerate fullerene structures is given. It is based on a topdown approach, and it is fast enough to generate, for example, all 1812 isomers of C 60 in less than 20 seconds on an SGIworkstation. The method described can easily be generalised for 3regular spherical maps with no face having more than 6 edges in its boundary.
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 9 (4 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.
Fast Backtracking Principles Applied to Find New Cages
 Ninth Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 1998
"... We describe how standard backtracking rules of thumb were successfully applied to the problem of characterizing (3; g) cages, the minimum order 3regular graphs of girth g. It took just 5 days of cpu time (compared to 259 days for previous authors) to verify the (3; 9)cages, and we were able to c ..."
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Cited by 9 (3 self)
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We describe how standard backtracking rules of thumb were successfully applied to the problem of characterizing (3; g) cages, the minimum order 3regular graphs of girth g. It took just 5 days of cpu time (compared to 259 days for previous authors) to verify the (3; 9)cages, and we were able to confirm that (3; 11)cages have order 112 for the first time ever. The lower bound for a (3; 13)cage is improved from 196 to 202 using the same approach. Also, we determined that a (3; 14)cage has order at least 258. 1 Cages In this paper, we consider finite undirected graphs. Any undefined notation follows Bondy and Murty [7]. The girth of a graph is the size of a smallest cycle. A (r; g) cage is an rregular graph of minimum order with girth g. It is known that (r; g)cages always exist [11]. Some nice pictures of small cages are given in [9, pp. 5458]. The classification of the cages has attracted much interest amongst the graph theory community, and many of these have special nam...
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"... We count all latin cubes of order n ≤ 6 and latin hypercubes of order n ≤ 5 and dimension d ≤ 5. We classify these (hyper)cubes into isotopy classes and paratopy classes (main classes). For the same values of n and d we classify all dary quasigroups of order n into isomorphism classes and also coun ..."
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We count all latin cubes of order n ≤ 6 and latin hypercubes of order n ≤ 5 and dimension d ≤ 5. We classify these (hyper)cubes into isotopy classes and paratopy classes (main classes). For the same values of n and d we classify all dary quasigroups of order n into isomorphism classes and also count them according to the number of identity elements they possess (meaning we have counted the dary loops). We also give an exact formula for the number of (isomorphism classes of) dary quasigroups of order 3 for every d. Then we give a number of constructions for dary quasigroups with a specific number of identity elements. In the process, we prove that no 3ary loop of order n can have exactly n − 1 identity elements (but no such result holds in dimensions other than 3). Finally, we give some new examples of latin cuboids which cannot be extended to latin cubes. 1 Basic definitions Let [n] denote the set {1, 2,..., n} and let [n] d denote the cartesian product [n] × [n] × · · · × [n] of d copies of [n]. By a hypercube of order n and dimension d we mean a ddimensional