Embedding Steiner triple systems in hexagon triple systems

A hexagon triple is the graph consisting of the three triangles (triples) {a,b,c},{c,d,e}, and {e,f,a}, where a,b,c,d,e, and f are distinct. The triple {a,c,e} is called an inside triple. A hexagon triple system of order n is a pair (X,H) where H is a collection of edge disjoint hexagon triples which partitions the edge set of K_n with vertex set X. The inside triples form a partial Steiner triple system. We show that any Steiner triple system of order n can be embedded in the inside triples of a hexagon triple system of order approximately 3n.     

Pasch trades with a negative block

A Steiner triple system of order v, STS(v), may be called equivalent to another STS(v) if one can be converted to the other by a sequence of three simple operations involving Pasch trades with a single negative block. It is conjectured that any two STS(v)s on the same base set are equivalent in this sense. We prove that the equivalence class containing a given system S on a base set V contains all the systems that can be obtained from S by any sequence of well over one hundred distinct trades, and that this equivalence class contains all isomorphic copies of S on V. We also show that there are trades which cannot be effected by means of Pasch trades with a single negative block.     

Perfect dexagon triple systems with given subsystems

The graph consisting of the six triples (or triangles) {a,b,c}, {c,d,e}, {e,f,a}, {x,a,y}, {x,c,z}, {x,e,w}, where a,b,c,d,e,f,x,y,z and w are distinct, is called a dexagon triple. In this case the six edges {a,c}, {c,e}, {e,a}, {x,a}, {x,c}, and {x,e} form a copy of K₄ and are called the inside edges of the dexagon triple. A dexagon triple system of order v is a pair (X,D), where D is a collection of edge disjoint dexagon triples which partitions the edge set of 3K_v. A dexagon triple system is said to be perfect if the inside copies of K₄ form a block design. In this note, we investigate the existence of a dexagon triple system with a subsystem. We show that the necessary conditions for the existence of a dexagon triple system of order v with a sub-dexagon triple system of order u are also sufficient.     

Minimum embedding of Steiner triple systems into -designs I

A (K₄-e)-design on v+w points embeds a Steiner triple system (STS) if there is a subset of v points on which the graphs of the design induce the blocks of a STS. It is established that wv/3, and that when equality is met that such a minimum embedding of an STS(v) exists, except when v=15.     

Doyen–Wilson theorem for perfect hexagon triple systems

The graph consisting of the three 3-cycles (or triples) (a,b,c), (c,d,e), and (e,f,a), where a,b,c,d,e and f are distinct is called a hexagon triple. The 3-cycle (a,c,e) is called an inside 3-cycle; and the 3-cycles (a,b,c), (c,d,e), and (e,f,a) are called outside 3-cycles. A hexagon triple system of order v is a pair (X,C), where C is a collection of edge disjoint hexagon triples which partitions the edge set of 3K_v. Note that the outside 3-cycles form a 3-fold triple system. If the hexagon triple system has the additional property that the collection of inside 3-cycles (a,c,e) is a Steiner triple system it is said to be perfect. In 2004, Küçükçifçi and Lindner had shown that there is a perfect hexagon triple system of order v if and only if and v≥7. In this paper, we investigate the existence of a perfect hexagon triple system with a given subsystem. We show that there exists a perfect hexagon triple system of order v with a perfect sub-hexagon triple system of order u if and only if v≥2u+1, and u≥7, which is a perfect hexagon triple system analogue of the Doyen–Wilson theorem.     

Steiner triple systems with two disjoint subsystems

It is shown that there exists a triangle decomposition of the graph obtained from the complete graph of order v by removing the edges of two vertex disjoint complete subgraphs of orders u and w if and only if u,w, and v are odd, (mod 3), and . Such decompositions are equivalent to group divisible designs with block size 3, one group of size u, one group of size w, and v – u – w groups of size 1. This result settles the existence problem for Steiner triple systems having two disjoint specified subsystems, thereby generalizing the well‐known theorem of Doyen and Wilson on the existence of Steiner triple systems with a single specified subsystem. © 2005 Wiley Periodicals, Inc. J Combin Designs     

New bounds for the broadcast domination number of a graph

A dominating broadcast on a graph G = (V, E) is a function f: V → {0, 1, ..., diam G} such that f(v) ≤ e(v) (the eccentricity of v) for all v ∈ V and such that each vertex is within distance f(v) from a vertex v with f(v) > 0. The cost of a broadcast f is σ(f) = Σ _v∈V f(v), and the broadcast number λ _b (G) is the minimum cost of a dominating broadcast. A set X ? V(G) is said to be irredundant if each x ∈ X dominates a vertex y that is not dominated by any other vertex in X; possibly y = x. The irredundance number ir (G) is the cardinality of a smallest maximal irredundant set of G. We prove the bound λ_b(G) ≤ 3 ir(G)/2 for any graph G and show that equality is possible for all even values of ir (G). We also consider broadcast domination as an integer programming problem, the dual of which provides a lower bound for λ_b.     

Representing Graphs in Steiner Triple Systems

Let G =  (V, E) be a simple graph and let T =  (P, B) be a Steiner triple system. Let φ be a one-to-one function from V to P. Any edge e =  {u, v} has its image {φ(u), φ(v)} in a unique block in B. We also denote this induced function from edges to blocks by φ. We say that T represents G if there exists a one-to-one function φ : V → P such that the induced function φ : E → B is also one-to-one; that is, if we can represent vertices of the graph by points of the triple system such that no two edges are represented by the same block. In this paper we examine when a graph can be represented by an STS. First, we find a bound which ensures that every graph of order n is represented in some STS of order f(n). Second, we find a bound which ensures that every graph of order n is represented in every STS of order g(n). Both of these answers are related to finding an independent set in an STS. Our question is a generalization of finding such independent sets. We next examine which graphs can be represented in STS’s of small orders. Finally, we give bounds on the orders of STS’s that are guaranteed to embed all graphs of a given maximum degree.     

On list coloring Steiner triple systems

We study the list chromatic number of Steiner triple systems. We show that for every integer s there exists n₀=n₀(s) such that every Steiner triple system on n points STS(n) with n≥n₀ has list chromatic number greater than s. We also show that the list chromatic number of a STS(n) is always within a log n factor of its chromatic number. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 314–322, 2009     

Disjoint Paths in Graphs II,A Special Case

Let G be a graph,{a,b,c} í V(G) \{a,b,c\}\subseteq V(G) , and {a￠,b￠,c￠} í V(G) \{a',b',c'\}\subseteq V(G) such that {a,b,c} 1 {a￠,b￠,c￠} \{a,b,c\}\neq \{a',b',c'\} . We say that (G,{a,c}, {a￠,c￠}, (b, b￠)) (G,\{a,c\}, \{a',c'\}, (b, b')) is an obstruction if, for any three vertex disjoint paths from {a, b, c} to {a', b', c'} in G, one path is from b to b'. In this paper, we characterize a special class of obstructions. This will be used to characterize all obstructions.     

Existence of non‐resolvable Steiner triple systems

We consider two well‐known constructions for Steiner triple systems. The first construction is recursive and uses an STS(v) to produce a non‐resolvable STS(2v + 1), for v ≡ 1 (mod 6). The other construction is the Wilson construction that we specify to give a non‐resolvable STS(v), for v ≡ 3 (mod 6), v > 9. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 16–24, 2005.     

On the Intersection Problem for Steiner Triple Systems of Different Orders

A Steiner triple system of order v, or STS(v), is a pair (V, ) with V a set of v points and a set of 3-subsets of V called blocks or triples, such that every pair of distinct elements of V occurs in exactly one triple. The intersection problem for STS is to determine the possible numbers of blocks common to two Steiner triple systems STS(u), (U, ), and STS(v), (V, ), with U⊆V. The case where U=V was solved by Lindner and Rosa in 1975. Here, we let U⊂V and completely solve this question for v−u=2,4 and for v≥2u−3. supported by NSERC research grant #OGP0170220. supported by NSERC postdoctoral fellowship. supported by NSERC research grant #OGP007621.     

On the acyclic choosability of graphs

A proper vertex coloring of a graph G = (V,E) is acyclic if G contains no bicolored cycle. A graph G is L‐list colorable if for a given list assignment L = {L(v): v ∈ V}, there exists a proper coloring c of G such that c (v) ∈ L(v) for all v ∈ V. If G is L‐list colorable for every list assignment with |L (v)| ≥ k for all v ∈ V, then G is said k‐choosable. A graph is said to be acyclically k‐choosable if the obtained coloring is acyclic. In this paper, we study the links between acyclic k‐choosability of G and Mad(G) defined as the maximum average degree of the subgraphs of G and give some observations about the relationship between acyclic coloring, choosability, and acyclic choosability. © 2005 Wiley Periodicals, Inc. J Graph Theory 51: 281–300, 2006     

Semiboolean SQS-skeins

We will present a counter example to the conjecture that the class of boolean SQS-skeins is defined by the equation q(x, u, q(y, u, z)) = q(q(x, u, y), u, z ). The SQS-skeins satisfying this equation will be seen to be exactly those SQS-skeins that correspond to Steiner quadruple systems whose derived Steiner triple systems are all projective geometries.     

Nonexistence of sparse triple systems over abelian groups and involutions

In 1973 Paul Erdős conjectured that there is an integer v ₀(r) such that, for every v>v ₀(r) and v≡1,3 (mod 6), there exists a Steiner triple system of order v, containing no i blocks on i+2 points for every 1<i≤r. Such an STS is said to be r-sparse. In this paper we consider relations of automorphisms of an STS to its sparseness. We show that for every r≥13 there exists no point-transitive r-sparse STS over an abelian group. This bound and the classification of transitive groups give further nonexistence results on block-transitive, flag-transitive, 2-transitive, and 2-homogeneous STSs with high sparseness. We also give stronger bounds on the sparseness of STSs having some particular automorphisms with small groups. As a corollary of these results, it is shown that various well-known automorphisms, such as cyclic, 1-rotational over arbitrary groups, and involutions, prevent an STS from being high-sparse.      

Steiner centers and Steiner medians of graphs

Let G be a connected graph and S a set of vertices of G. The Steiner distance of S is the smallest number of edges in a connected subgraph of G that contains S and is denoted by d_G(S) or d(S). The Steiner n-eccentricity e_n(v) and Steiner n-distance d_n(v) of a vertex v in G are defined as e_n(v)=max{d(S)| SV(G), |S|=n and vS} and d_n(v)=∑{d(S)| SV(G), |S|=n and vS}, respectively. The Steiner n-center C_n(G) of G is the subgraph induced by the vertices of minimum n-eccentricity. The Steiner n-median M_n(G) of G is the subgraph induced by those vertices with minimum Steiner n-distance. Let T be a tree. Oellermann and Tian [O.R. Oellermann, S. Tian, Steiner centers in graphs, J. Graph Theory 14 (1990) 585–597] showed that C_n(T) is contained in C_n+1(T) for all n2. Beineke et al. [L.W. Beineke, O.R. Oellermann, R.E. Pippert, On the Steiner median of a tree, Discrete Appl. Math. 68 (1996) 249–258] showed that M_n(T) is contained in M_n+1(T) for all n2. Then, Oellermann [O.R. Oellermann, On Steiner centers and Steiner medians of graphs, Networks 34 (1999) 258–263] asked whether these containment relationships hold for general graphs. In this note we show that for every n2 there is an infinite family of block graphs G for which C_n(G)C_n+1(G). We also show that for each n2 there is a distance–hereditary graph G such that M_n(G)M_n+1(G). Despite these negative examples, we prove that if G is a block graph then M_n(G) is contained in M_n+1(G) for all n2. Further, a linear time algorithm for finding the Steiner n-median of a block graph is presented and an efficient algorithm for finding the Steiner n-distances of all vertices in a block graph is described.     

Disjoint Paths in Graphs I,¶3-planar Graphs and Basic Obstructions

Let G be a graph,{a,b,c} í V(G) \{a,b,c\}\subseteq V(G) , and {a￠,b￠,c￠} í V(G) \{a',b',c'\}\subseteq V(G) such that {a,b,c} 1 {a￠,b￠,c￠} \{a,b,c\}\neq \{a',b',c'\} . We say that (G,{a,c}, {a￠,c￠}, (b, b￠)) (G,\{a,c\}, \{a',c'\}, (b, b')) is an obstruction if, for any three vertex disjoint paths from {a, b, c} to {a', b', c'} in G, one path is from b to b'. Robertson and Seymour asked the problem of characterizing all obstructions. In this paper, we present a list of "basic" obstructions and show how to produce other obstructions from these basic ones. We also prove results about disjoint paths in graphs. Results in this paper will be used in subsequent papers to characterize all obstructions.     

The Fine Intersection Problem for Steiner Triple Systems

The intersection of two Steiner triple systems and is the set . 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 satisfies and . 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). Received: January 23, 2006. Final Version received: September 2, 2006     

On the Binary Codes of Steiner Triple Systems

The binary code spanned by the rows of the point byblock incidence matrix of a Steiner triple system STS(v)is studied. A sufficient condition for such a code to containa unique equivalence class of STS(v)'s of maximalrank within the code is proved. The code of the classical Steinertriple system defined by the lines in PG(n-1,2)(n3), or AG(n,3) (n3) is shown to contain exactly v codewordsof weight r=(v-1)/2, hence the system is characterizedby its code. In addition, the code of the projective STS(2ⁿ-1)is characterized as the unique (up to equivalence) binary linearcode with the given parameters and weight distribution. In general,the number of STS(v)'s contained in the code dependson the geometry of the codewords of weight r. Itis demonstrated that the ovals and hyperovals of the definingSTS(v) play a crucial role in this geometry. Thisrelation is utilized for the construction of some infinite classesof Steiner triple systems without ovals.     

Quasi‐embeddings of Steiner triple systems,or Steiner triple systems of different orders with maximum intersection

In this paper, we present a conjecture that is a common generalization of the Doyen–Wilson Theorem and Lindner and Rosa's intersection theorem for Steiner triple systems. Given u, v ≡ 1,3 (mod 6), u < v < 2u + 1, we ask for the minimum r such that there exists a Steiner triple system such that some partial system can be completed to an STS , where |?| = r. In other words, in order to “quasi‐embed” an STS(u) into an STS(v), we must remove r blocks from the small system, and this r is the least such with this property. One can also view the quantity (u(u ? 1)/6) ? r as the maximum intersection of an STS(u) and an STS(v) with u < v. We conjecture that the necessary minimum r = (v ? u) (2u + 1 ? v)/6 can be achieved, except when u = 6t + 1 and v = 6t + 3, in which case it is r = 3t for t ≠ 2, or r = 7 when t = 2. Using small examples and recursion, we solve the cases v ? u = 2 and 4, asymptotically solve the cases v ? u = 6, 8, and 10, and further show for given v ? u > 2 that an asymptotic solution exists if solutions exist for a run of consecutive values of u (whose required length is no more than v ? u). Some results are obtained for v close to 2u + 1 as well. The cases where ≈ 3u/2 seem to be the hardest. © 2004 Wiley Periodicals, Inc.