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A pure Mendelsohn triple system of order v, denoted by PMTS( v), is a pair $$(X,\mathcal {B})$$ where X is a v-set and $$\mathcal {B}$$ is a collection of cyclic triples on X such that every ordered pair of X belongs to exactly one triple of $$\mathcal {B}$$ and if $$\langle a,b,c\rangle \in \mathcal {B}$$ implies $$\langle c,b,a\rangle \notin \mathcal {B}$$ . An overlarge set of PMTS( v), denoted by OLPMTS( v), is a collection $$\{(Y{\setminus }\{y_i\},{\mathcal {A}}_i)\}_i$$ , where Y is a $$(v+1)$$ -set, $$y_i\in Y$$ , each $$(Y{\setminus }\{y_i\},{\mathcal {A}}_i)$$ is a PMTS( v) and these $${\mathcal {A}}_i$$ s form a partition of all cyclic triples on Y. It is shown in [] that there exists an OLPMTS( v) for $$v\equiv 1,3$$ (mod 6), $$v>3$$ , or $$v \equiv 0,4$$ (mod 12). In this paper, we shall discuss the existence problem of OLPMTS( v) s for $$v\equiv 6,10$$ (mod 12) and get the following conclusion: there exists an OLPMTS( v) if and only if $$v\equiv 0,1$$ (mod 3), $$v>3$$ and $$v\ne 6$$ . [ABSTRACT FROM AUTHOR]

A complete weighted graph, $$G(X,\varGamma ,W)$$ , is considered. Let $$\tilde{X}\subset X$$ be some subset of vertices and, by definition, a Steiner tree is any tree in the graph G such that the set of the tree vertices includes set $$\tilde{X}$$ . The Steiner tree problem consists of constructing the minimum-length Steiner tree in graph G, for a given subset of vertices $$\tilde{X}$$ The effectively computable estimate of the minimal Steiner tree is obtained in terms of the mean value and the variance over the set of all Steiner trees. It is shown that in the space of the lengths of the graph edges, there exist some regions where the obtained estimate is better than the minimal spanning tree-based one. [ABSTRACT FROM AUTHOR]

Let P be a point set on the plane, and consider whether P is quadrangulatable, that is, whether there exists a 2-connected plane graph G with each edge a straight segment such that V( G) = P, that the outer cycle of G coincides with the convex hull Conv( P) of P, and that each finite face of G is quadrilateral. It is easy to see that it is possible if and only if an even number of points of P lie on Conv( P). Hence we give a k-coloring to P, and consider the same problem, avoiding edges joining two vertices of P with the same color. In this case, we always assume that the number of points of P lying on Conv( P) is even and that any two consecutive points on Conv( P) have distinct colors. However, for every k ≥ 2, there is a k-colored non-quadrangulatable point set P. So we introduce Steiner points, which can be put in any position of the interior of Conv( P) and each of which may be colored by any of the k colors. When k = 2, Alvarez et al. proved that if a point set P on the plane consists of $${\frac{n}{2}}$$ red and $${\frac{n}{2}}$$ blue points in general position, then adding Steiner points Q with $${|Q| \leq \lfloor \frac{n-2}{6} \rfloor + \lfloor \frac{n}{4} \rfloor +1}$$ , P ∪ Q is quadrangulatable, but there exists a non-quadrangulatable 3-colored point set for which no matter how many Steiner points are added. In this paper, we define the winding number for a 3-colored point set P, and prove that a 3-colored point set P in general position with a finite set Q of Steiner points added is quadrangulatable if and only if the winding number of P is zero. When P ∪ Q is quadrangulatable, we prove $${|Q| \leq \frac{7n+34m-48}{18}}$$ , where | P| = n and the number of points of P in Conv( P) is 2 m. [ABSTRACT FROM AUTHOR]

For a block design $${\mathcal{D}}$$ , a series of block intersection graphs G, or i-BIG( $${\mathcal{D}}$$), i = 0, . . . , k is defined in which the vertices are the blocks of $${\mathcal{D}}$$ , with two vertices adjacent if and only if the corresponding blocks intersect in exactly i elements. A silver graph G is defined with respect to a maximum independent set of G, called an α- set. Let G be an r-regular graph and c be a proper ( r + 1)-coloring of G. A vertex x in G is said to be rainbow with respect to c if every color appears in the closed neighborhood $${N[x] = N(x) \cup \{x\}}$$ . Given an α-set I of G, a coloring c is said to be silver with respect to I if every $${x\in I}$$ is rainbow with respect to c. We say G is silver if it admits a silver coloring with respect to some I. Finding silver graphs is of interest, for a motivation and progress in silver graphs see Ghebleh et al. (Graphs Combin 24(5):429-442, ) and Mahdian and Mahmoodian (Bull Inst Combin Appl 28:48-54, ). We investigate conditions for 0-BIG( $${\mathcal{D}}$$) and 1-BIG( $${\mathcal{D}}$$) of Steiner 2-designs $${{\mathcal{D}}=S(2,k,v)}$$ to be silver. [ABSTRACT FROM AUTHOR]

Given a complete metric space X and a compact set $${C\subset X}$$ , the famous Steiner (or minimal connection) problem is that of finding a set S of minimum length (one-dimensional Hausdorff measure $${\mathcal H^1)}$$) among the class of sets In this paper we provide conditions on existence of minimizers and study topological regularity results for solutions of this problem. We also study the relationships between several similar variants of the Steiner problem. At last, we provide some applications to locally minimal sets. [ABSTRACT FROM AUTHOR]

In this article we prove almost sure convergence, in the L distance, of sequences of random Steiner symmetrizations of measurable sets having finite measure to the ball having the same measure. From this result we deduce analogous statements concerning the almost sure convergence to the spherical symmetrization of random Steiner symmetrizations of non negative L functions in the natural norm and uniform convergence of non negative continuous functions with bounded support. The latter result is finally used to prove that sequences of random symmetrizations of a compact set converge almost surely in the Hausdorff distance to the ball having the same measure, providing another proof of Mani-Levitska's conjecture besides the one given in 2006 by Van Schaftingen (Topol Methods Nonlinear Anal 28(1): 61-85, ). [ABSTRACT FROM AUTHOR]

A family ( X, B), ( X, B), ..., ( X, B) of q STS( υ)s is a λ-fold large set of STS( υ) and denoted by LSTS( υ) if every 3-subset of X is contained in exactly λ STS( υ)s of the collection. It is indecomposable and denoted by IDLSTS( υ) if there does not exist an LSTS( υ) contained in the collection for any λ′ < λ. In this paper, we show that for λ = 5, 6, there is an IDLSTS( υ) for υ ≡ 1 or 3 (mod 6) with the exception IDLSTS(7). [ABSTRACT FROM AUTHOR]

The concepts of edge Steiner set and edge Steiner number of a graph are investigated in this study. A necessary and sufficient condition for a graph G to satisfy ste(G) = |V (G)|-1, where ste(G) denotes the edge Steiner number of G, is obtained. Edge Steiner sets in the joins of graphs are also studied and the Steiner numbers of these graphs are determined. [ABSTRACT FROM AUTHOR]

We give a construction that produces 6-sparse Steiner triple systems of order v for all sufficiently large v of the form 3 p, p prime and p ≡ 3 (mod 4). We also give a complete list of all 429 6-sparse systems with v < 10000 produced by this construction. [ABSTRACT FROM AUTHOR]

The intersection of two Steiner triple systems $$(X,{\mathcal{A}})$$ and $$(X,{\mathcal{B}})$$ is the set $${\mathcal{A}}\cap{\mathcal{B}}$$ . The fine intersection problem for Steiner triple systems is to determine for each v, the set I( v), consisting of all possible pairs ( m, n) such that there exist two Steiner triple systems of order v whose intersection $${\mathcal{I}}$$ satisfies $$|\cup_{A\in{\mathcal{I}}} A|=m$$ and $$|{\mathcal{I}}|=n$$ . We show that for v ≡ 1 or 3 (mod 6), | I( v)| = Θ( v 3), where previous results only imply that | I( v)| = Ω( v 2). [ABSTRACT FROM AUTHOR]