Page 269 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 269
COMBINATORIAL DESIGNS (MS-16)
Bipartite 2-factorizations of complete multigraphs via layering
Mateja Šajna, msajna@uottawa.ca
University of Ottawa, Canada
Coauthor: Amin Bahmanian
Layering is in principle a simple method that allows us to obtain a type-specific 2-factorization
of a complete multigraph (or complete multigraph minus a 1-factor) from existing 2-factoriza-
tions of complete multigraphs and complete multigraphs minus a 1-factor. This technique is
particularly effective when constructing bipartite 2-factorizations; that is, 2-factorizations with
all cycles of even length.
In this talk, we shall give a thorough introduction to layering, and then describe new bipar-
tite 2-factorizations of complete multigraphs obtained by layering. In particular, for complete
multigraphs and bipartite 2-factors with no 2-cycles, we obtain a complete solution to the Ober-
wolfach Problem and an almost complete solution to the Hamilton-Waterloo Problem.
This is joint work with Amin Bahmanian.
On some periodic Golay pairs and pairwise balanced designs
Andrea Švob, asvob@math.uniri.hr
Department of Mathematics, University of Rijeka, Croatia
Coauthors: Ronan Egan, Dean Crnkovic´, Doris Dumicˇic´ Danilovic´
In this talk we will show how a relationship between certain pairwise balanced designs with
v points and periodic Golay pairs of length v can be useful to construct periodic Golay pairs.
The talk is based on recent work [1] where we construct pairwise balanced designs with v
points under specific block conditions having an assumed cyclic automorphism group, and using
isomorph rejection which is compatible with equivalence of corresponding periodic Golay pairs,
we complete a classification of periodic Golay pairs of length less than 40, up to equivalence.
Further, we will show how we use similar tools to construct new periodic Golay pairs of lengths
greater than 40 where classifications remain incomplete, and demonstrate that under some extra
conditions on its automorphism group, a periodic Golay pair of length 90 do not exist.
References
[1] D. Crnkovic´, D. Dumicˇic´ Danilovic´, R. Egan, A. Švob, Periodic Golay pairs and pairwise
balanced designs, J. Algebraic Combin., to appear.
Dual incidences and t-designs in elementary abelian groups
Kristijan Tabak, kxtcad@rit.edu
Rochester Institute of Technology - Croatia, Croatia
Let q be a prime and Eqn is an elementary abelian group of order qn. Let H be a collection of
some subgroups of Eqn of order qk. A pair (Eqn, H) is a t − (n, k, λ)q design if every subgroup
of Eqn of order qt is contained in exactly λ groups from H. This definition corresponds to the
classical definition of a q-analog design.
267
Bipartite 2-factorizations of complete multigraphs via layering
Mateja Šajna, msajna@uottawa.ca
University of Ottawa, Canada
Coauthor: Amin Bahmanian
Layering is in principle a simple method that allows us to obtain a type-specific 2-factorization
of a complete multigraph (or complete multigraph minus a 1-factor) from existing 2-factoriza-
tions of complete multigraphs and complete multigraphs minus a 1-factor. This technique is
particularly effective when constructing bipartite 2-factorizations; that is, 2-factorizations with
all cycles of even length.
In this talk, we shall give a thorough introduction to layering, and then describe new bipar-
tite 2-factorizations of complete multigraphs obtained by layering. In particular, for complete
multigraphs and bipartite 2-factors with no 2-cycles, we obtain a complete solution to the Ober-
wolfach Problem and an almost complete solution to the Hamilton-Waterloo Problem.
This is joint work with Amin Bahmanian.
On some periodic Golay pairs and pairwise balanced designs
Andrea Švob, asvob@math.uniri.hr
Department of Mathematics, University of Rijeka, Croatia
Coauthors: Ronan Egan, Dean Crnkovic´, Doris Dumicˇic´ Danilovic´
In this talk we will show how a relationship between certain pairwise balanced designs with
v points and periodic Golay pairs of length v can be useful to construct periodic Golay pairs.
The talk is based on recent work [1] where we construct pairwise balanced designs with v
points under specific block conditions having an assumed cyclic automorphism group, and using
isomorph rejection which is compatible with equivalence of corresponding periodic Golay pairs,
we complete a classification of periodic Golay pairs of length less than 40, up to equivalence.
Further, we will show how we use similar tools to construct new periodic Golay pairs of lengths
greater than 40 where classifications remain incomplete, and demonstrate that under some extra
conditions on its automorphism group, a periodic Golay pair of length 90 do not exist.
References
[1] D. Crnkovic´, D. Dumicˇic´ Danilovic´, R. Egan, A. Švob, Periodic Golay pairs and pairwise
balanced designs, J. Algebraic Combin., to appear.
Dual incidences and t-designs in elementary abelian groups
Kristijan Tabak, kxtcad@rit.edu
Rochester Institute of Technology - Croatia, Croatia
Let q be a prime and Eqn is an elementary abelian group of order qn. Let H be a collection of
some subgroups of Eqn of order qk. A pair (Eqn, H) is a t − (n, k, λ)q design if every subgroup
of Eqn of order qt is contained in exactly λ groups from H. This definition corresponds to the
classical definition of a q-analog design.
267
