Mathematical and algorithmic methods for finding disjoint Rosa-type sequences

A Rosa sequence of order n is a sequence S = (s1; s2; ..., s2n+1) of 2n + 1 integers satisfying the conditions: (1) for every k ∈ {1; 2;...; n} there are exactly two elements sᵢ; sj ∈ S such that si = sj = k; (2) if sᵢ = sj = k; i < j, then j - i = k; and (3) sn+1 = 0 (sn+1 is called the hook). Two Rosa sequences S and S' are disjoint if sᵢ = sj = k = s't = s'ᵤ implies that {i;j} ≠ {t,u}, for all k = 1;..., n. In 2014, Linek, Mor, and Shalaby [18] introduced several new constructions for Skolem, hooked Skolem, and Rosa rectangles. In this thesis, we gave new constructions for four mutually disjoint hooked Rosa sequences and we used them to generate cyclic triple systems CTS₄(v). We also obtained new constructions for two disjoint m-fold Skolem sequences, two disjoint m-fold Rosa sequences, and two disjoint indecomposable 2-fold Rosa sequences of order n. Again, we can use these sequences to construct cyclic 2-fold 3-group divisible design 3-GDD and disjoint cyclically indecomposable CTS₄(6n+3). Finally, we introduced exhaustive search algorithms to find all distinct hooked Rosa sequences, as well as maximal and maximum disjoint subsets of (hooked) Rosa sequences

On the number of additive permutations and Skolem-type sequences

Cavenagh and Wanless recently proved that, for sufficiently large odd n, the number of transversals in the Latin square formed from the addition table for integers modulo n is greater than (3.246)n. We adapt their proof to show that for sufficiently large t the number of additive permutations on [-t,t] is greater than (3.246)2t+1 and we go on to derive some much improved lower bounds on the numbers of Skolem-type sequences. For example, it is shown that for sufficiently large t ≡ 0$ or 3 (mod 4), the number of split Skolem sequences of order n=7t+3 is greater than (3.246)6t+3. This compares with the previous best bound of 2⌊n/3⌋

Starter sequences: generalizations and applications

In this thesis we introduce new types of starter sequences, pseudo-starter sequences, starter-labellings, and generalized (extended) starter sequences. We apply these new sequences to graph labeling. All the necessary conditions for the existence of starter, pseudo-starter, extended, m-fold, excess, and generalized (extended) starter sequences are determined, and some of these conditions are shown to be sufficient. The relationship between starter sequences and graph labellings is introduced. Moreover, the starter-labeling and the minimum hooked starter-labeling of paths, cycles, and k- windmills are investigated. We show that all paths, cycles, and k-windmills can be starter-labelled or minimum starter-labelled

Disjoint skolem-type sequences and applications

Let D = {i₁, i₂,..., in} be a set of n positive integers. A Skolem-type sequence of order n is a sequence of i such that every i ∈ D appears exactly twice in the sequence at position aᵢ and bᵢ, and |bᵢ - aᵢ| = i. These sequences might contain empty positions, which are filled with 0 elements and called hooks. For example, (2; 4; 2; 0; 3; 4; 0; 3) is a Skolem-type sequence of order n = 3, D = f2; 3; 4g and two hooks. If D = f1; 2; 3; 4g we have (1; 1; 4; 2; 3; 2; 4; 3), which is a Skolem-type sequence of order 4 and zero hooks, or a Skolem sequence. In this thesis we introduce additional disjoint Skolem-type sequences of order n such as disjoint (hooked) near-Skolem sequences and (hooked) Langford sequences. We present several tables of constructions that are disjoint with known constructions and prove that our constructions yield Skolem-type sequences. We also discuss the necessity and sufficiency for the existence of Skolem-type sequences of order n where n is positive integers

Cyclic packing designs and simple cyclic leaves constructed from Skolem-type sequences

A Packing Design, or a PD(v; k; λ) is a pair (V; β) where V is a v-set of points and β is a set of k-subsets (blocks) such that any 2-subset of V appears in at most λ blocks. PD(v; k; λ) is cyclic if its automorphism group contains a v-cycle, and it is called a cyclic packing design. The edges in the multigraph λKᵥ not contained in the packing form the leaves of the CPD(v; k; λ); denoted by leave (v; k; λ) : In 2012, Silvesan and Shalaby used Skolem-type sequences to provide a complete proof for the existence of cyclic BIBD(v; 3; λ) for all admissible orders v and λ. In this thesis, we use Skolem-type sequences to find all cyclic packing designs with block size 3 for a cyclic BIBD(v; 3; λ) and find the spectrum of leaves graph of the cyclic packing designs, for all admissible orders v and λ with the optimal leaves, as well as determine the number of base blocks for every λ when k = 3

Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)

International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference “Algebras, graphs and ordered set” (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Maurice’s many scientific interests:• Lattices and ordered sets• Combinatorics and graph theory• Set theory and theory of relations• Universal algebra and multiple valued logic• Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum