Page 271 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 271
COMBINATORIAL DESIGNS (MS-16)
McGuire, Tonchev and Ward proved that indeed the 2-rank of a unital on 28 points is between
19 and 27 and that there is a unique 2-(28, 4, 1) design, the Ree unital R(3), with 2-rank 19. In
the same year, Jaffe and Tonchev showed that there is no unital on 28 points of 2-rank 20 and
there are exactly 4 isomorphism classes of unitals of rank 21.
Here, we present the complete classification by computer of unitals of 2-rank 22, 23 and 24.
There are 12 isomorphism classes of unitals of 2-rank 22, 78 isomorphism classes of unitals of
2-rank 23, and 298 isomorphism classes of unitals of 2-rank 24.
Heffter Arrays and Biembbedings of Cycle Systems on Orientable
Surfaces
E. S¸ ule Yazıcı, eyazici@ku.edu.tr
Koç University, Turkey
Coauthors: Diane Donovan, Kevin Burrage, Nicholas Cavenagh
In this talk we will review the recent developments on square Heffter arrays, H(n; k), and their
applications on face 2-colourable embeddings of the complete graph K2nk+1 on an orientable
surfaces.
Square Heffter arrays, H(n; k), are n×n arrays such that each row and each column contains
k filled cells, each row and column sum is divisible by 2nk + 1 and either x or −x appears in
the array for each integer 1 ≤ x ≤ nk.
Archdeacon noted that a Heffter array, satisfying two additional conditions, yields a face
2-colourable embedding of the complete graph K2nk+1 on an orientable surface, where for each
colour, the faces give a k-cycle system. These necessary conditions pertain to cyclic orderings of
the entries in each row and each column of the Heffter array and are: (1) for each row and each
column the sequential partial sums determined by the cyclic ordering must be distinct modulo
2nk + 1; (2) the composition of the cyclic orderings of the rows and columns is equivalent to a
single cycle permutation on the entries in the array.
We construct Heffter arrays that satisfy condition (1) whenever (a) k ≡ 0 mod 4; or (b)
n ≡ 1 mod 4 and k ≡ 3 mod 4; or (c) n ≡ 0 mod 4, k ≡ 3 mod 4 and n k. As a corollary
to the above we obtain pairs of orthogonal k-cycle decompositions of K2nk+1.
Furthermore we study when these arrays satisfy condition (2). We show the existence of face
2-colourable embeddings of cycle decompositions of the complete graph when n ≡ 1 mod 4
and k ≡ 3 mod 4, n k ≥ 7 (provided that when n ≡ 0 mod 3 then k ≡ 7 mod 12).
Block designs constructed from orbit matrices using a modified genetic
algorithm
Tin Zrinski, tin.zrinski@math.uniri.hr
Department of Mathematics, University of Rijeka, Croatia
Genetic algorithms (GA) are search and optimization heuristic population-based methods which
are inspired by the natural evolution process. In this talk, we will present a method of con-
structing incidence matrices of block designs combining the method of construction with orbit
matrices and a modified genetic algorithm. With this method we managed to find some new
non-isomorphic S(2,5,45) designs.
269
McGuire, Tonchev and Ward proved that indeed the 2-rank of a unital on 28 points is between
19 and 27 and that there is a unique 2-(28, 4, 1) design, the Ree unital R(3), with 2-rank 19. In
the same year, Jaffe and Tonchev showed that there is no unital on 28 points of 2-rank 20 and
there are exactly 4 isomorphism classes of unitals of rank 21.
Here, we present the complete classification by computer of unitals of 2-rank 22, 23 and 24.
There are 12 isomorphism classes of unitals of 2-rank 22, 78 isomorphism classes of unitals of
2-rank 23, and 298 isomorphism classes of unitals of 2-rank 24.
Heffter Arrays and Biembbedings of Cycle Systems on Orientable
Surfaces
E. S¸ ule Yazıcı, eyazici@ku.edu.tr
Koç University, Turkey
Coauthors: Diane Donovan, Kevin Burrage, Nicholas Cavenagh
In this talk we will review the recent developments on square Heffter arrays, H(n; k), and their
applications on face 2-colourable embeddings of the complete graph K2nk+1 on an orientable
surfaces.
Square Heffter arrays, H(n; k), are n×n arrays such that each row and each column contains
k filled cells, each row and column sum is divisible by 2nk + 1 and either x or −x appears in
the array for each integer 1 ≤ x ≤ nk.
Archdeacon noted that a Heffter array, satisfying two additional conditions, yields a face
2-colourable embedding of the complete graph K2nk+1 on an orientable surface, where for each
colour, the faces give a k-cycle system. These necessary conditions pertain to cyclic orderings of
the entries in each row and each column of the Heffter array and are: (1) for each row and each
column the sequential partial sums determined by the cyclic ordering must be distinct modulo
2nk + 1; (2) the composition of the cyclic orderings of the rows and columns is equivalent to a
single cycle permutation on the entries in the array.
We construct Heffter arrays that satisfy condition (1) whenever (a) k ≡ 0 mod 4; or (b)
n ≡ 1 mod 4 and k ≡ 3 mod 4; or (c) n ≡ 0 mod 4, k ≡ 3 mod 4 and n k. As a corollary
to the above we obtain pairs of orthogonal k-cycle decompositions of K2nk+1.
Furthermore we study when these arrays satisfy condition (2). We show the existence of face
2-colourable embeddings of cycle decompositions of the complete graph when n ≡ 1 mod 4
and k ≡ 3 mod 4, n k ≥ 7 (provided that when n ≡ 0 mod 3 then k ≡ 7 mod 12).
Block designs constructed from orbit matrices using a modified genetic
algorithm
Tin Zrinski, tin.zrinski@math.uniri.hr
Department of Mathematics, University of Rijeka, Croatia
Genetic algorithms (GA) are search and optimization heuristic population-based methods which
are inspired by the natural evolution process. In this talk, we will present a method of con-
structing incidence matrices of block designs combining the method of construction with orbit
matrices and a modified genetic algorithm. With this method we managed to find some new
non-isomorphic S(2,5,45) designs.
269
