Cliques in Steiner systems (original) (raw)

Strong Ramsey theorems for Steiner systems

Transactions of the American Mathematical Society, 1987

It is shown that the class of partial Steiner (fc, Z)-systems has the edge Ramsey property, i.e., we prove that for every partial Steiner (k, i)-system Q there exists a partial Steiner (fc, Z)-system)i such that for every partition of the edges of H into two classes one can find an induced monochromatic copy of Q. As an application we get that the class of all graphs without cycles of lengths 3 and 4 has the edge Ramsey property. This solves a longstanding problem in the area.

A Construction of Almost Steiner Systems

Journal of Combinatorial Designs, 2013

Let n, k, and t be integers satisfying n > k > t ≥ 2. A Steiner system with parameters t, k, and n is a k-uniform hypergraph on n vertices in which every set of t distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for t ≥ 6. In this note we prove that for every k > t ≥ 2 and sufficiently large n, there exists an almost Steiner system with parameters t, k, and n; that is, there exists a k-uniform hypergraph on n vertices such that every set of t distinct vertices is covered by either one or two edges.

On the Independence Number of Steiner Systems

Combinatorics, Probability and Computing, 2013

A partial Steiner (n,r,l)-system is an r-uniform hypergraph on n vertices in which every set of l vertices is contained in at most one edge. A partial Steiner (n,r,l)-system is complete if every set of l vertices is contained in exactly one edge. In a hypergraph , the independence number α() denotes the maximum size of a set of vertices in containing no edge. In this article we prove the following. Given integers r,l such that r ≥ 2l − 1 ≥ 3, we prove that there exists a partial Steiner (n,r,l)-system such that \alpha(\HH) \lesssim \biggl(\frac{l-1}{r-1}(r)_l\biggr)^{\frac{1}{r-1}}n^{\frac{r-l}{r-1}} (\log n)^{\frac{1}{r-1}} \quad \mbox{ as }n \rightarrow \infty.$$ This improves earlier results of Phelps and Rödl, and Rödl and Ŝinajová. We conjecture that it is best possible as it matches the independence number of a random r-uniform hypergraph of the same density. If l = 2 or l = 3, then for infinitely many r the partial Steiner systems constructed are complete for infinitely man...

Graph-theoretic perspective on a special class of Steiner Systems

Eprint Arxiv 1410 5855, 2014

We study S(t−1,t,2t)S(t-1,t,2t)S(t−1,t,2t), which is a special class of Steiner systems. Explicit constructions for designing such systems are developed under a graph-theoretic platform where Steiner systems are represented in the form of uniform hypergraphs. The constructions devised are then used to study the 222-coloring properties of these uniform hypergraphs.

The Forcing Restrained Steiner Number of a Graph

International journal of engineering and advanced technology, 2019

A restrained Steiner set of a connected graph of order ≥ is a set ⊆ ()such that is a Steiner set, and if either = or the subgraph [ − ] inducedby [ − ] has no isolated vertices. The restrained Steiner number of isthe minimum cardinality of its restrained Steiner sets and any restrained Steinerset of cardinality is a minimum restrained Steiner set of. For a minimum restrained Steiner set of , a subset ⊆ is called a forcing subset for if is the unique minimum restrained Steiner set containing. A forcing subset for of minimum cardinality is a minimum forcing subset of. The forcing restrained Steiner number of , denoted by , is the cardinality of a minimum forcingsubset of. The forcing restrained Steiner number of , denoted by is = { }, where the minimum is taken over all minimum restrainedSteiner sets in. Some general properties satisfied by the concept forcing restrained Steiner number are studied. The forcing restrained Steiner number of certain classes of graphs is determined. It is shown that for every pair , ofintegers with ≤ < and ≥ , there exists a connected graph such that = and = .

The Forcing Edge Steiner Number of a Graph

International journal of pure and applied mathematics, 2018

For a connected graph G = (V, E), a set W ⊆ V (G) is called an edge Steiner set of G if every edge of G is contained in a Steiner W-tree of G. The edge Steiner number s1(G) of G is the minimum cardinality of its edge Steiner sets and any edge Steiner set of cardinality s1(G) is a minimum edge Steiner set of G. For a minimum edge Steiner set W of G, a subset T ⊆ W is called a forcing subset for W if W is the unique minimum edge Steiner set containing T. A forcing subset for W of minimum cardinality is a minimum forcing subset of W. The forcing edge Steiner number of W , denoted by f s1(W), is the cardinality of a minimum forcing subset of W. The forcing edge Steiner number of G, denoted by f s1(G), is f s1(G) = min{f s1(W)}, where the minimum is taken over all minimum edge Steiner sets W in G. Some general properties satisfied by this concept are studied. The forcing edge Steiner numbers of certain classes of graphs are determined. It is shown for every pair of integers with 0 ≤ a ≤ b, b ≥ 2 and b − a − 1 > 0, there exists a connected graph G such that f s1(G) = a and s1(G) = b.

On a More General Characterisation of Steiner Systems

Designs, Codes and Cryptography, 2005

We introduce [k,d]-sparse geometries of cardinality n, which are natural generalizations of partial Steiner systems PS(t,k;n), with d=2(k−t+1). We will verify whether Steiner systems are characterised in the following way. (*) Let Gamma=(mathcalP,mathcalB)\Gamma=(\mathcal{P},\mathcal{B})Gamma=(mathcalP,mathcalB) be a [k,2(k−t+1)]-sparse geometry of cardinality n, with fracn+12>k>t>1\frac{n+1}{2} \> k \> t \> 1fracn+12>k>t>1 . If ∣mathcalB∣genchooset/kchooset|\mathcal{B}| \ge {n \choose t}/{k \choose t}∣mathcalB∣genchooset/kchooset , then Γ is a S(t,k;n). If (*) holds for fixed parameters t, k and n, then we say S(t,k;n) satisfies, or has, characterisation (*). We could not prove (*) in general, but we prove the Theorems 1, 2, 3 and 4, which state conditions under which (*) is satisfied. Moreover, we verify characterisation (*) for every Steiner system appearing in list of the sporadic Steiner systems of small cardinality, and the list of infinite series of Steiner systems, both mentioned in the latest edition of the book ‘Design Theory’ by T. Beth, D. Jungnickel and H. Lenz. As an interesting application, one can use these results to build (almost) maximal binary codes in the following way. Every [k,d]-sparse geometry is associated with a [k,d]-sparse binary code of the same size (let mathcalP=p1,ldots,pn\mathcal{P} = \{ p_1, \ldots, p_n \}mathcalP=p1,ldots,pn and link every block BinmathcalBB \in \mathcal{B}BinmathcalB with the code word (ci)1leilen(c_i)_{1 \le i \le n}(ci)1leilen where c i =1 if and only if the point p i is a member of B), so one can construct maximal [k,d]-sparse binary codes using (partial) Steiner systems. These [k,d]-sparse codes can then be used as building bricks for binary codes having a bigger variety of weights (the weight of a code word is the sum of its entries).

THE SHARPE BOUND STEINER NUMBER IN SOME CLASSES OF GRAPHS

IAEME Publications, 2017

For a connected graph G=(V,E) of order at least 3 and a nonempty subset   the minimum size of a connected subgraph containing , if the subgraph a tree with the distance  then the tree is called Steiner   . A set    is called Steiner set of  if every vertex of  is contained in a Steiner W-tree of . The Minimum cardinality of its Steiner set called the Steiner number denoted as . We present some classes of graphs for which Steiner numberis known. We have estimated the Sharpe bound for the Steiner number of some classes of graphs such as wheel  Fan , KmKn,, ,     .

A new proof of the Fisher-Ryan bounds for the number of cliques of a graph

2000

Tutte and Nash-Williams, independently, gave necessary and sufficient conditions for a connected graph to have at least t edgedisjoint spanning trees. Gusfield introduced the concept of edgetoughness η(G) of a connected graph G, defined as the minimum |S|/(ω(G − S) − 1) taken over all edge-disconnecting sets S of G, where ω(G − S) is the number of connected components of G − S. If a graph has edge-toughness η(G), Tutte and Nash-Williams's theorem says that the maximum number of edge-disjoint spanning trees of a graphs is given by η(G). Kundu used this result to show that a graph with edge-connectivity λ(G) has at least λ(G)/2 edgedisjoint spanning trees. In this paper we investigate to which extent the above results can be generalized to a graph G = (V, E) with a distinguished subset of vertices K. We obtain lower bounds for the maximum number of edge-disjoint Steiner trees of G (minimal trees of G containing K) in terms of λK (G), the K-edge-connectivity of G (defined as the minimum number of edges whose removal disconnects K). In [3] we introduced the K-edge-toughness of a graph, ηK (G) (which coincides with η(G) when K = V). We extent some of the properties of the edge-toughness of a graph to the K-edge-toughness and we show by mean of a counterexample that the maximum number of disjoint Steiner trees can be less than ηK (G) when K = V. We conclude with some conjectures regarding these bounds.