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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⌋
Langford sequences and a product of digraphs
Skolem and Langford sequences and their many generalizations have
applications in numerous areas. The -product is a generalization of
the direct product of digraphs. In this paper we use the -product
and super edge-magic digraphs to construct an exponential number of Langford
sequences with certain order and defect. We also apply this procedure to
extended Skolem sequences.Comment: 10 pages, 6 figures, to appear in European Journal of Combinatoric
Distance labelings: a generalization of Langford sequences
A Langford sequence of order m and defect d can be identified with a labeling of the vertices of a path of order 2m in which each label from d up to d + m − 1 appears twice and in which the vertices that have been labeled with k are at distance k. In this paper, we introduce two generalizations of this labeling that are related to distances. The basic idea is to assign nonnegative integers to vertices in such a way that if n vertices (n > 1) have been labeled with k then they are mutually at distance k. We study these labelings for some well known families of graphs. We also study the existence of these labelings in general. Finally, given a sequence or a set of nonnegative integers, we study the existence of graphs that can be labeled according to this sequence or set.The research conducted in this document by the first author has been supported by the
Spanish Research Council under project MTM2011-28800-C02-01 and symbolically by the
Catalan Research Council under grant 2014SGR1147
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
Wythoff Wisdom
International audienceSix authors tell their stories from their encounters with the famous combinatorial game Wythoff Nim and its sequences, including a short survey on exactly covering systems
UNIFORM THREE-CLASS REGULAR PARTIAL STEINER TRIPLE SYSTEMS WITH UNIFORM DEGREES
A Partial Steiner Triple system (X, T) is a finite set of points C and a collection T of 3-element subsets of C that every pair of points intersect in at most 1 triple. A 3-class regular PSTS (3-PSTS) is a PSTS where the points can be partitioned into 3 classes (each class having size m, n and p respectively) such that no triple belongs to any class and any two points from the same class occur in the same number of triples (a, b and c respectively). The 3-PSTS is said to be uniform if m = n = p. In this thesis, we have mostly focused on the existence of uniform 3-PSTS with uniform degrees (a = b = c)
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