Page 264 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 264
COMBINATORIAL DESIGNS (MS-16)
Equitably 2-colourable even cycle systems
Francesca Merola, merola@mat.uniroma3.it
Università Roma Tre, Italy
Coauthor: Andrea C. Burgess
An -cycle decomposition of a graph G is said to be equitably c-colourable if there is a c-vertex-
colouring of G such that each colour is represented (approximately) an equal number of times
on each cycle: more precisely, we ask that in each cycle C of the decomposition, each colour
appears on /c or /c of the vertices of C. In this talk, we consider the case c = 2 and
present some new results on the existence of 2-colourable even -cycle systems of the cocktail
party graph Kv − I. In particular, we determine a complete existence result for equitably 2-
colourable -cycle decompositions of Kv − I, even, in the cases that v ≡ 0, 2 (mod ), or
is a power of 2, or ∈ {2q, 4q} for q an odd prime power, or ≤ 30. We will also discuss some
work in progress on analogous problems for cycles of odd length.
Weakly self-orthogonal designs and related linear codes
Vedrana Mikulic´ Crnkovic´, vmikulic@math.uniri.hr
University of Rijeka, Croatia
Coauthor: Ivona Traunkar
A 1-design is weakly self-orthogonal if all the block intersection numbers have the same parity.
If both k and the block intersection numbers are even then 1-design is called self-orthogonal and
its incidence matrix generates a self-orthogonal code. We analyze extensions, of the incidence
matrix and an orbit matrix of a weakly self-orthogonal 1-design, that generates a self-orthogonal
code over the finite field. Additionally, we develop methods for constructing LCD codes by
extending the incidence matrix and an orbit matrix of a weakly self-orthogonal 1-design.
The classification of 2-(v, k, λ) designs, with λ > 1 and (r, λ) = 1,
admitting a flag-transitive automorphism group
Alessandro Montinaro, alessandro.montinaro@unisalento.it
University of Salento, Italy
A classical subject in Theory of Designs is the study of 2-designs admitting an automorphism
group fulfilling prescribed properties. Within this research area, it is of great interest the study
of 2-(v, k, λ) designs D admitting a flag-transitive automorphism group G. Since they have
been classified for λ = 1 and G AΓL1(q) by Buekenhout et al. (1990), a special attention is
devoted to the general case λ > 1. In this setting, a first natural generalization of the case λ = 1
is represented by λ > 1 and gcd(r, λ) = 1, where r is the replication number of D. Then G acts
point-primitively on D by a result of Dembowski (1968), and Soc(G), the socle of G, is either
an elementary abelian p-group for some prime p, or a non abelian simple group by a result of
Zeischang (1988). Starting from these two results, such 2-designs have been recently classified
for G AΓL1(q) by Biliotti et al. and by Alavi, Zhou et al. according to whether Soc(G) is an
elementary abelian p-group or a non abelian simple group, respectively.
The aim of the talk is to survey the classification of 2-(v, k, λ) designs D, with λ > 1
262
Equitably 2-colourable even cycle systems
Francesca Merola, merola@mat.uniroma3.it
Università Roma Tre, Italy
Coauthor: Andrea C. Burgess
An -cycle decomposition of a graph G is said to be equitably c-colourable if there is a c-vertex-
colouring of G such that each colour is represented (approximately) an equal number of times
on each cycle: more precisely, we ask that in each cycle C of the decomposition, each colour
appears on /c or /c of the vertices of C. In this talk, we consider the case c = 2 and
present some new results on the existence of 2-colourable even -cycle systems of the cocktail
party graph Kv − I. In particular, we determine a complete existence result for equitably 2-
colourable -cycle decompositions of Kv − I, even, in the cases that v ≡ 0, 2 (mod ), or
is a power of 2, or ∈ {2q, 4q} for q an odd prime power, or ≤ 30. We will also discuss some
work in progress on analogous problems for cycles of odd length.
Weakly self-orthogonal designs and related linear codes
Vedrana Mikulic´ Crnkovic´, vmikulic@math.uniri.hr
University of Rijeka, Croatia
Coauthor: Ivona Traunkar
A 1-design is weakly self-orthogonal if all the block intersection numbers have the same parity.
If both k and the block intersection numbers are even then 1-design is called self-orthogonal and
its incidence matrix generates a self-orthogonal code. We analyze extensions, of the incidence
matrix and an orbit matrix of a weakly self-orthogonal 1-design, that generates a self-orthogonal
code over the finite field. Additionally, we develop methods for constructing LCD codes by
extending the incidence matrix and an orbit matrix of a weakly self-orthogonal 1-design.
The classification of 2-(v, k, λ) designs, with λ > 1 and (r, λ) = 1,
admitting a flag-transitive automorphism group
Alessandro Montinaro, alessandro.montinaro@unisalento.it
University of Salento, Italy
A classical subject in Theory of Designs is the study of 2-designs admitting an automorphism
group fulfilling prescribed properties. Within this research area, it is of great interest the study
of 2-(v, k, λ) designs D admitting a flag-transitive automorphism group G. Since they have
been classified for λ = 1 and G AΓL1(q) by Buekenhout et al. (1990), a special attention is
devoted to the general case λ > 1. In this setting, a first natural generalization of the case λ = 1
is represented by λ > 1 and gcd(r, λ) = 1, where r is the replication number of D. Then G acts
point-primitively on D by a result of Dembowski (1968), and Soc(G), the socle of G, is either
an elementary abelian p-group for some prime p, or a non abelian simple group by a result of
Zeischang (1988). Starting from these two results, such 2-designs have been recently classified
for G AΓL1(q) by Biliotti et al. and by Alavi, Zhou et al. according to whether Soc(G) is an
elementary abelian p-group or a non abelian simple group, respectively.
The aim of the talk is to survey the classification of 2-(v, k, λ) designs D, with λ > 1
262
