Search

A partial Room square is maximal if no further pair of elements can be placed into any unoccupied cell without violating the conditions that define a partial Room square. This article is concerned with determining the spectrum of volumes of maximal partial Room squares of order n where the volume is the number of occupied cells and the order n is even. [ABSTRACT FROM AUTHOR]

In this article, we attempt to survey all that has been written about 6-cycle systems. In addition, several new results, including many enumeration results, are included here for the first time. Our survey concludes with a list of open problems. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex-set V = V(G) and edge-set E = E(G). A 1-factor of G (also called perfect matching) is a factor of G of degree 1, that is, a set of pairwise disjoint edges which partitions V. A 1-factorization of G is a partition of its edge-set E into 1-factors. For a graph G to have a 1-factor, |V(G)| must be even, and for a graph G to admit a 1-factorization, G must be regular of degree r, 1 ≤ r ≤ |V| − 1. One can find in the literature at least two extensive surveys [69] and [89] and also a whole book [90] devoted to 1-factorizations of (mainly) complete graphs. A 1-factorization of G is said to be perfect if the union of any two of its distinct 1-factors is a Hamiltonian cycle of G. An early survey on perfect 1-factorizations (abbreviated as P1F) of complete graphs is [83]. In the book [90] a whole chapter (Chapter 16) is devoted to perfect 1-factorizations of complete graphs. It is the purpose of this article to present what is known to-date on P1Fs, not only of complete graphs but also of other regular graphs, primarily cubic graphs. [ABSTRACT FROM AUTHOR]

Copyright of MEDISAN is the property of Centro Provincial de Informacion en Ciencias Medicas de Santiago de Cuba and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We prove that an almost $$2$$ -perfect $$6$$ -cycle system of order $$n$$ exists if and only if $$n \equiv 1$$ or $$9\ (mod\ 12)$$ , and that an almost $$2$$ -perfect maximum packing with $$6$$ -cycles of order $$n$$ exists for all $$n \ge 6$$ . [ABSTRACT FROM AUTHOR]

Squashed 6-cycle systems are introduced as a natural counterpart to 2-perfect 6-cycle systems. The spectrum for the latter has been determined previously in [5]. We determine completely the spectrum for squashed 6-cycle systems, and also for squashed 6-cycle packings. [ABSTRACT FROM AUTHOR]

It is well known that when [ABSTRACT FROM AUTHOR]

A mitre in a Steiner triple system is a set of five triples on seven points, in which two are disjoint. Recursive constructions for Steiner triple systems containing no mitre are developed, leading to such anti-mitre systems for at least 9/16 of the admissible orders. Computational results for small cyclic Steiner triple systems are also included. [ABSTRACT FROM AUTHOR]

A c-hybrid triple system of order v is a decomposition of the complete v-vertex digraph into c cyclic tournaments of order 3 and $$\frac{{v(v - 1)}}{3} - c$$ transitive tournaments of order 3. Hybrid triple systems generalize directed triple systems ( c = 0) and Mendelsohn triple systems ( c = v(v − 1)/3); omitting directions yields an underlying twofold triple system. The spectrum of v and c for which a c-hybrid triple system of order v exists is completely determined in this paper. Using (cubic) block intersection graphs, we then show that every twofold triple system of order $$v\left( {having b_v = \frac{{v(v - 1)}}{3}blocks} \right)$$ underlies a c-hybrid triple system with $$c \geqslant \frac{{2b_v }}{3}$$ . Examples are constructed for all sufficiently large v, for which this maximum is at most $$\left( {\frac{7}{{10}} + \varepsilon } \right)b_v $$ . The lower bound here is proved by establishing bounds on F( n, r), the size of minimum cardinality vertex feedback sets in n-vertex i-connected cubic multigraphs having r repeated edges. We establish that $$F_0 (n,r) \leqslant \frac{n}{2},$$ , $$F_1 (n,r) \leqslant \frac{{3n}}{8} + \frac{r}{4} + \frac{1}{2}, and F_2 (n,r) \leqslant \frac{{(n + r)}}{3}for n > 8$$ . These bounds are all tight, and the latter is used to derive the lower bound in the design theoretic problem. [ABSTRACT FROM AUTHOR]

Every graph having vertex degrees zero and two satisfying the basic necessary conditions is the leave of a maximal partial triple system, with one exception ( C ⋃ C). The proof technique is direct, using the method of differences. [ABSTRACT FROM AUTHOR]