Page 270 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 270
COMBINATORIAL DESIGNS (MS-16)
We introduce two incidence structures denoted by Dmax and Dmin with H as set of points.
The blocks of Dmax are labeled by maximal subgroups of Eqn, while the blocks of Dmin are
labeled by groups of order q.
We fully describe a duality between Dmax and Dmin by proving some identities over group
rings. The proven results are used to provide a full description of incidence matrices of Dmax
and Dmin and their mutual dependance.
On the Oberwolfach Problem for single-flip 2-factors via graceful
labelings
Tommaso Traetta, tommaso.traetta@unibs.it
Università degli Studi di Brescia, Italy
Coauthors: Peter Danziger, Andrea C. Burgess
The Oberwolfach Problem, posed by Ringel in 1967 and still open, asks for each odd integer
v > 1 and each 2-regular graph F of order v to determine whether there is a decomposition of
the complete graph Kv into copies of F .
We construct solutions whenever F has a sufficiently large odd cycle meeting a specified
lower bound and, in addition, F has a single-flip automorphism (i.e. an involutory automor-
phism acting as a reflection on exactly one cycle). For even orders v, we give analogous results
for the maximum packing and minimum covering variants of the problem. We also show a
similar result when the edges of Kv have multiplicity 2, but in this case we only require that F
has a sufficiently large cycle.
Our methods build on the techniques used in [2] and involve a doubling construction de-
fined in [1] which we apply to graceful labelings of 2-regular graphs with a vertex removed,
allowing us to explicitly construct solutions to the Oberwolfach Problem with well-behaved
automorphisms.
This is joint work with Andrea Burgess and Peter Danziger.
References
[1] M. Buratti, T. Traetta. 2-starters, graceful labelings, and a doubling construction for the
Oberwolfach Problem. J. Combin. Des. 20 (2012), 483–503.
[2] T. Traetta. A complete solution to the two-table Oberwolfach Problems. J. Combin. Theory
Ser. A 120 (2013), 984–997.
On the classification of unitals on 28 points of low rank
Alfred Wassermann, alfred.wassermann@uni-bayreuth.de
University of Bayreuth, Germany
Coauthor: Vladimir Tonchev
Unitals are combinatorial 2-(q3 + 1, q + 1, 1) designs. In 1981, Brouwer constructed 138 non-
isomorphic unitals for q = 3, i.e. 2-(28, 4, 1) designs. He observed that the 2-rank of the
constructed unitals is at least 19, where the p-rank of a design is defined as the rank of the
incidence matrix between points and blocks of the design over the finite field GF(p). In 1998,
268
We introduce two incidence structures denoted by Dmax and Dmin with H as set of points.
The blocks of Dmax are labeled by maximal subgroups of Eqn, while the blocks of Dmin are
labeled by groups of order q.
We fully describe a duality between Dmax and Dmin by proving some identities over group
rings. The proven results are used to provide a full description of incidence matrices of Dmax
and Dmin and their mutual dependance.
On the Oberwolfach Problem for single-flip 2-factors via graceful
labelings
Tommaso Traetta, tommaso.traetta@unibs.it
Università degli Studi di Brescia, Italy
Coauthors: Peter Danziger, Andrea C. Burgess
The Oberwolfach Problem, posed by Ringel in 1967 and still open, asks for each odd integer
v > 1 and each 2-regular graph F of order v to determine whether there is a decomposition of
the complete graph Kv into copies of F .
We construct solutions whenever F has a sufficiently large odd cycle meeting a specified
lower bound and, in addition, F has a single-flip automorphism (i.e. an involutory automor-
phism acting as a reflection on exactly one cycle). For even orders v, we give analogous results
for the maximum packing and minimum covering variants of the problem. We also show a
similar result when the edges of Kv have multiplicity 2, but in this case we only require that F
has a sufficiently large cycle.
Our methods build on the techniques used in [2] and involve a doubling construction de-
fined in [1] which we apply to graceful labelings of 2-regular graphs with a vertex removed,
allowing us to explicitly construct solutions to the Oberwolfach Problem with well-behaved
automorphisms.
This is joint work with Andrea Burgess and Peter Danziger.
References
[1] M. Buratti, T. Traetta. 2-starters, graceful labelings, and a doubling construction for the
Oberwolfach Problem. J. Combin. Des. 20 (2012), 483–503.
[2] T. Traetta. A complete solution to the two-table Oberwolfach Problems. J. Combin. Theory
Ser. A 120 (2013), 984–997.
On the classification of unitals on 28 points of low rank
Alfred Wassermann, alfred.wassermann@uni-bayreuth.de
University of Bayreuth, Germany
Coauthor: Vladimir Tonchev
Unitals are combinatorial 2-(q3 + 1, q + 1, 1) designs. In 1981, Brouwer constructed 138 non-
isomorphic unitals for q = 3, i.e. 2-(28, 4, 1) designs. He observed that the 2-rank of the
constructed unitals is at least 19, where the p-rank of a design is defined as the rank of the
incidence matrix between points and blocks of the design over the finite field GF(p). In 1998,
268
