Page 267 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 267
COMBINATORIAL DESIGNS (MS-16)
References
[1] M. Buratti, A. Pasotti, T. Traetta, A reduction of the spectrum problem for odd sun systems
and the prime case, J. Combin. Des. 29 (2021), 5–37.
[2] A. Pasotti, T. Traetta, Even sun systems of the complete graph, in preparation.
Regular 1-factorizations of complete graphs with orthogonal spanning
trees
Gloria Rinaldi, gloria.rinaldi@unimore.it
University of Modena and Reggio Emilia, Italy
A 1-factorization F of a complete graph K2n is said to be G-regular if G is an automorphism
group of F acting sharply transitively on the vertex-set. The problem of determining which
groups can realize such a situation dates back to a result by Hartman and Rosa (1985) which
solved the problem when G is a cyclic group. It is also well known that this problem simplifies
somewhat when n is odd: G must be the semi-direct product of Z2 with its normal comple-
ment and G always realizes a 1−factorization of K2n upon which it acts sharply transitively
on vertices. When n is even the problem is still open, even though several classes of groups
were tested in the recent past. An attempt to obtain a fairly precise description of groups and
1-factorizations satisfying this symmetry constraint could be done by imposing further condi-
tions. For example some non existence results were achieved by assuming the existence of a
1-factor fixed by the action of the group, further results were obtained when the number of fixed
1−factors is as large as possible. In this talk we focus our attention on the possibility of con-
structing G-regular 1-factorizations of K2n together with a complete set of isomorphic spanning
trees orthogonal to the 1−factorization. Here orthogonal tree means that the tree shares exactly
one edge with each 1−factor. We see how to realize such a situation when n is odd and examine
some classes of groups in the case n even.
A few new triplanes
Sanja Rukavina, sanjar@math.uniri.hr
University of Rijeka, Croatia
Coauthor: Dean Crnkovic´
An incidence structure D = (P, B, I), with point set P, block set B and incidence I is a
t-(v, k, λ) design, if |P| = v, every block B ∈ B is incident with precisely k points, and
every t distinct points are together incident with precisely λ blocks. We consider triplanes, i.e.
symmetric block designs with λ = 3. Triplanes of order 12, i.e. symmetric (71,15,3) designs,
have the greatest number of points among all known triplanes and it is not known if a triplane
(v, k, 3) exists for v > 71.
In this talk, in addition to reviewing previously known results, we give the first example of
a triplane of order 12 that doesn’t admit an automorphism of order 3.
265
References
[1] M. Buratti, A. Pasotti, T. Traetta, A reduction of the spectrum problem for odd sun systems
and the prime case, J. Combin. Des. 29 (2021), 5–37.
[2] A. Pasotti, T. Traetta, Even sun systems of the complete graph, in preparation.
Regular 1-factorizations of complete graphs with orthogonal spanning
trees
Gloria Rinaldi, gloria.rinaldi@unimore.it
University of Modena and Reggio Emilia, Italy
A 1-factorization F of a complete graph K2n is said to be G-regular if G is an automorphism
group of F acting sharply transitively on the vertex-set. The problem of determining which
groups can realize such a situation dates back to a result by Hartman and Rosa (1985) which
solved the problem when G is a cyclic group. It is also well known that this problem simplifies
somewhat when n is odd: G must be the semi-direct product of Z2 with its normal comple-
ment and G always realizes a 1−factorization of K2n upon which it acts sharply transitively
on vertices. When n is even the problem is still open, even though several classes of groups
were tested in the recent past. An attempt to obtain a fairly precise description of groups and
1-factorizations satisfying this symmetry constraint could be done by imposing further condi-
tions. For example some non existence results were achieved by assuming the existence of a
1-factor fixed by the action of the group, further results were obtained when the number of fixed
1−factors is as large as possible. In this talk we focus our attention on the possibility of con-
structing G-regular 1-factorizations of K2n together with a complete set of isomorphic spanning
trees orthogonal to the 1−factorization. Here orthogonal tree means that the tree shares exactly
one edge with each 1−factor. We see how to realize such a situation when n is odd and examine
some classes of groups in the case n even.
A few new triplanes
Sanja Rukavina, sanjar@math.uniri.hr
University of Rijeka, Croatia
Coauthor: Dean Crnkovic´
An incidence structure D = (P, B, I), with point set P, block set B and incidence I is a
t-(v, k, λ) design, if |P| = v, every block B ∈ B is incident with precisely k points, and
every t distinct points are together incident with precisely λ blocks. We consider triplanes, i.e.
symmetric block designs with λ = 3. Triplanes of order 12, i.e. symmetric (71,15,3) designs,
have the greatest number of points among all known triplanes and it is not known if a triplane
(v, k, 3) exists for v > 71.
In this talk, in addition to reviewing previously known results, we give the first example of
a triplane of order 12 that doesn’t admit an automorphism of order 3.
265
