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Constructions for Cubic Graphs With Large Girth
 Electronic Journal of Combinatorics
, 1998
"... The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a welldefined integer ¯ 0 (g), the smallest number of vertices for which a cubic graph with girth at least g exists, and furthermore, the minimum value ¯ 0 (g) is attained by a ..."
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Cited by 35 (0 self)
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The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a welldefined integer ¯ 0 (g), the smallest number of vertices for which a cubic graph with girth at least g exists, and furthermore, the minimum value ¯ 0 (g) is attained by a graph whose girth is exactly g. The values of ¯ 0 (g) when 3 g 8 have been known for over thirty years. For these values of g each minimal graph is unique and, apart from the case g = 7, a simple lower bound is attained. This paper is mainly concerned with what happens when g 9, where the situation is quite different. Here it is known that the simple lower bound is attained if and only if g = 12. A number of techniques are described, with emphasis on the construction of families of graphs fG i g for which the number of vertices n i and the girth g i are such that n i 2 cg i for some finite constant c. The optimum value of c is known to lie between 0:5 and 0:75. At the end of the p...
Moore graphs and beyond: A survey of the degree/diameter problem
 ELECTRONIC JOURNAL OF COMBINATORICS
, 2013
"... The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds – called Moore bounds – for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bo ..."
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Cited by 26 (4 self)
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The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds – called Moore bounds – for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem ‘from above’, remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem ‘from below’. This survey aims to give an overview of the current stateoftheart of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moorelike bounds for special types of graphs and digraphs, such as vertextransitive, Cayley, planar, bipartite, and many others, on
Algorithms for Greechie Diagrams
 Int. J. Theor. Phys
, 2000
"... Abstract. We give a new algorithm for generating Greechie diagrams with arbitrary chosen number of atoms or blocks (with 2,3,4,... atoms) and provide a computer program for generating the diagrams. The results show that the previous algorithm does not produce every diagram and that it is at least 10 ..."
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Cited by 7 (6 self)
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Abstract. We give a new algorithm for generating Greechie diagrams with arbitrary chosen number of atoms or blocks (with 2,3,4,... atoms) and provide a computer program for generating the diagrams. The results show that the previous algorithm does not produce every diagram and that it is at least 10 5 times slower. We also provide an algorithm and programs for checking of Greechie diagram passage by equations defining varieties of orthomodular lattices and give examples from Hilbert lattices. At the end we discuss some additional characteristics of Greechie diagrams. PACS numbers: 03.65.Bz, 02.10.By, 02.10.Gd
IsomorphFree Exhaustive Generation of Greechie Diagrams and Automated Checking of Their Passage by Orthomodular Lattice Equations
 Int. J. Theor. Phys
, 2000
"... Abstract. We give a new algorithm for generating Greechie diagrams with arbitrary chosen number of atoms or blocks (with 2,3,4,... atoms) and provide a computer program for generating the diagrams. The results show that the previous algorithm does not produce every diagram and that it is at least 10 ..."
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Cited by 1 (1 self)
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Abstract. We give a new algorithm for generating Greechie diagrams with arbitrary chosen number of atoms or blocks (with 2,3,4,... atoms) and provide a computer program for generating the diagrams. The results show that the previous algorithm does not produce every diagram and that it is at least 10 5 times slower. We also provide an algorithm and programs for checking of Greechie diagram passage by equations defining varieties of orthomodular lattices and give examples from Hilbert lattices. At the end we discuss some additional characteristics of Greechie diagrams. PACS numbers: 03.65.Bz, 02.10.By, 02.10.Gd
On the Classification of Resolvable 2(12, 6, 5c) Designs
"... In this paper we describe a backtrack search over parallel classes with a partial isomorph rejection to classify resolvable 2(12, 6, 5c) designs. We use the intersection pattern between the parallel classes and the fact that any resolvable 2(12, 6, 5c) design is also a resolvable 3(12, 6, 2c) des ..."
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In this paper we describe a backtrack search over parallel classes with a partial isomorph rejection to classify resolvable 2(12, 6, 5c) designs. We use the intersection pattern between the parallel classes and the fact that any resolvable 2(12, 6, 5c) design is also a resolvable 3(12, 6, 2c) design to effectively guide the search. The method was able to enumerate all nonsimple resolutions and a subfamily of simple resolutions of a 2(12, 6, 15) design. The method is also used to confirm the computer classification of the resolvable 2(12, 6, 5c) designs for c ∈ {1, 2}. A consistency checking based on the principle of double counting is used to verify the computation results.