2006
DOI: 10.1007/s00454-005-1224-9
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Small Triangle-Free Configurations of Points and Lines

Abstract: In the paper we show that all combinatorial triangle-free configurations for v ≤ 18 are geometrically realizable. We also show that there is a unique smallest astral (18 3 ) triangle-free configuration and its Levi graph is the generalized Petersen graph G(18, 5). In addition, we present geometric realizations of the unique flag transitive triangle-free configuration (20 3 ) and the unique point transitive triangle-free configuration (21 3 ).

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Cited by 24 publications
(33 citation statements)
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“…Our algorithm was tested for correctness against the known values of nonisomorphic combinatorial (v 4 ) and (v 3 ) configurations. In particular, we confirm the known counting of 971 171 combinatorial (18 4 ) configurations, (see [1], [2] and [3]), which lacked an independent verification. Let C be the combinatorial (19 4 ) configuration defined by block set given in Table 2.…”
Section: Computational Searchmentioning
confidence: 49%
“…Our algorithm was tested for correctness against the known values of nonisomorphic combinatorial (v 4 ) and (v 3 ) configurations. In particular, we confirm the known counting of 971 171 combinatorial (18 4 ) configurations, (see [1], [2] and [3]), which lacked an independent verification. Let C be the combinatorial (19 4 ) configuration defined by block set given in Table 2.…”
Section: Computational Searchmentioning
confidence: 49%
“…However, there exist other bicirculants with girth greater than 6. The corresponding configurations have been investigated in [3,1]. One way of extending this study is on the one hand to consider splittability of these more general bicirculants and on the other hand to study tricirculants [12], tetracirculants and beyond [7].…”
Section: Resultsmentioning
confidence: 99%
“…Let us first provide a construction to obtain a geometric (n k ) configuration for any k. We start with an unbalanced (k 1 , 1 k ) configuration, denoted G (1) k , that consists of a single line containing k points. Let G (i) k be a configuration that is obtained from…”
Section: Splittable Geometric (N K ) Configurationsmentioning
confidence: 99%
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“…In this sense many open problems in symmetric configurations, such as, for example, open problems on self-dual, point-and line-transitive (v 3 ) configurations is, through the above mentioned correspondence, are special cases of open problems on cubic vertextransitive graphs (see [19,20,22,34,58,92,94,103]). And similarly, open problems concerning weakly flag-transitive configurations are special cases of open problems on halfarc-transitive graphs (see [16,96,93]).…”
Section: Problemmentioning
confidence: 99%