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25
Nonorientable biembeddings of Steiner triple systems
 DISCRETE MATHEMATICS
, 2004
"... Constructions due to Ringel show that there exists a nonorientable face 2colourable triangular embedding of the complete graph on n vertices (equivalently a nonorientable biembedding of two Steiner triple systems of order n) for all n ≡ 3 (mod 6) with n ≥ 9. We prove the corresponding existence th ..."
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Cited by 7 (7 self)
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Constructions due to Ringel show that there exists a nonorientable face 2colourable triangular embedding of the complete graph on n vertices (equivalently a nonorientable biembedding of two Steiner triple systems of order n) for all n ≡ 3 (mod 6) with n ≥ 9. We prove the corresponding existence theorem for n ≡ 1 (mod 6) with n ≥ 13.
On Asymmetric Coverings and Covering Numbers
 J. COMBIN. DES
, 2003
"... An asymmetric covering D(n, d) is a collection of special subsets S of an nset such that every subset T of the nset is contained in at least one S with S  T  # d. In this paper we compute the smallest size of any D(n, 1) for n # 8. We also investigate "continuous" and " ..."
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Cited by 6 (0 self)
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An asymmetric covering D(n, d) is a collection of special subsets S of an nset such that every subset T of the nset is contained in at least one S with S  T  # d. In this paper we compute the smallest size of any D(n, 1) for n # 8. We also investigate "continuous" and "banded" versions of the problem. The latter involves the classical covering numbers C(n, k, k  1), and we determine the following new values: C(10, 5, 4) = 51, C(11, 7, 6, ) = 84, C(12, 8, 7) = 126, C(13, 9, 8) = 185 and C(14, 10, 9) = 259. We also find the number of nonisomorphic minimal covering designs in several cases.
The Steiner Quadruple Systems of Order 16
"... The Steiner quadruple systems of order 16 are classified up to isomorphism by means of an exhaustive computer search. The number of isomorphism classes of such designs is 1,054,163. Properties of the designs—including the orders of the automorphism groups and the structures of the derived Steiner t ..."
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Cited by 6 (2 self)
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The Steiner quadruple systems of order 16 are classified up to isomorphism by means of an exhaustive computer search. The number of isomorphism classes of such designs is 1,054,163. Properties of the designs—including the orders of the automorphism groups and the structures of the derived Steiner triple systems of order 15—are tabulated. A doublecounting consistency check is carried out to gain confidence in the correctness of the classification.
Reconstructing Extended Perfect Binary OneErrorCorrecting Codes from Their Minimum Distance Graphs
, 2008
"... The minimum distance graph of a code has the codewords as vertices and edges exactly when the Hamming distance between two codewords equals the minimum distance of the code. A constructive proof for reconstructibility of an extended perfect binary oneerrorcorrecting code from its minimum distance ..."
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Cited by 5 (3 self)
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The minimum distance graph of a code has the codewords as vertices and edges exactly when the Hamming distance between two codewords equals the minimum distance of the code. A constructive proof for reconstructibility of an extended perfect binary oneerrorcorrecting code from its minimum distance graph is presented. Consequently, inequivalent such codes have nonisomorphic minimum distance graphs. Moreover, it is shown that the automorphism group of a minimum distance graph is isomorphic to that of the corresponding code.
Steiner Triple Systems of Order 19 and 21 with Subsystems of Order 7
"... Steiner triple systems (STSs) with subsystems of order 7 are classi ed. For order 19, this classi cation is complete, but for order 21 it is restricted to Wilsontype systems, which contain three subsystems of order 7 on disjoint point sets. The classi ed STSs of order 21 are tested for resolv ..."
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Cited by 4 (3 self)
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Steiner triple systems (STSs) with subsystems of order 7 are classi ed. For order 19, this classi cation is complete, but for order 21 it is restricted to Wilsontype systems, which contain three subsystems of order 7 on disjoint point sets. The classi ed STSs of order 21 are tested for resolvability; none of them is doubly resolvable.
Properties of the Steiner triple systems of order 19
, 2010
"... Properties of the 11 084 874 829 Steiner triple systems of order 19 are examined. In particular, there is exactly one 5sparse, but no 6sparse, STS(19); there is exactly one uniform STS(19); there are exactly two STS(19) with no almost parallel classes; all STS(19) have chromatic number 3; all have ..."
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Cited by 4 (1 self)
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Properties of the 11 084 874 829 Steiner triple systems of order 19 are examined. In particular, there is exactly one 5sparse, but no 6sparse, STS(19); there is exactly one uniform STS(19); there are exactly two STS(19) with no almost parallel classes; all STS(19) have chromatic number 3; all have chromatic index 10, except for 4 075 designs with chromatic index 11 and two with chromatic index 12; all are 3resolvable; and there are exactly two 3existentially closed STS(19).
Orientable selfembeddings of Steiner triple systems of order 15
 ACTA MATH. UNIV. COMENIANAE
"... It is shown that for 78 of the 80 isomorphism classes of Steiner triple systems of order 15 it is possible to find a face 2colourable triangulation of the complete graphK15 in an orientable surface in which the colour classes both form representatives of the specified isomorphism class. For one o ..."
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It is shown that for 78 of the 80 isomorphism classes of Steiner triple systems of order 15 it is possible to find a face 2colourable triangulation of the complete graphK15 in an orientable surface in which the colour classes both form representatives of the specified isomorphism class. For one of the two remaining isomorphism classes it is proved that this is not possible. We also discuss the remaining open case.
Onefactorizations of regular graphs of order 12
 THE ELECTRONIC JOURNAL OF COMBINATORICS
, 2005
"... Algorithms for classifying onefactorizations of regular graphs are studied. The smallest open case is currently graphs of order 12; onefactorizations of rregular graphs of order 12 are here classified for r ≤ 6 and r = 10, 11. Two different approaches are used for regular graphs of small degree; ..."
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Algorithms for classifying onefactorizations of regular graphs are studied. The smallest open case is currently graphs of order 12; onefactorizations of rregular graphs of order 12 are here classified for r ≤ 6 and r = 10, 11. Two different approaches are used for regular graphs of small degree; these proceed onefactor by onefactor and vertex by vertex, respectively. For degree r = 11, we have onefactorizations of K12. These have earlier been classified, but a new approach is presented which views these as certain triple systems on 4n − 1 points and utilizes an approach developed for classifying Steiner triple systems. Some properties of the classified onefactorizations are also tabulated.