Existential closure of block intersection graphs of infinite designs having finite block size and index

In this article we study the n‐existential closure property of the block intersection graphs of infinite t‐(v, k, λ) designs for which the block size k and the index λ are both finite. We show that such block intersection graphs are 2‐e.c. when 2?t?k ? 1. When λ = 1 and 2?t?k, then a necessary and sufficient condition on n for the block intersection graph to be n‐e.c. is that n?min{t, ?(k ? 1)/(t ? 1)? + 1}. If λ?2 then we show that the block intersection graph is not n‐e.c. for any n?min{t + 1, ?k/t? + 1}, and that for 3?n?min{t, ?k/t?} the block intersection graph is potentially but not necessarily n‐e.c. The cases t = 1 and t = k are also discussed. © 2011 Wiley Periodicals, Inc. J Combin Designs 19: 85–94, 2011     

More on block intersection polynomials and new applications to graphs and block designs

The concept of intersection numbers of order r for t-designs is generalized to graphs and to block designs which are not necessarily t-designs. These intersection numbers satisfy certain integer linear equations involving binomial coefficients, and information on the non-negative integer solutions to these equations can be obtained using the block intersection polynomials introduced by P.J. Cameron and the present author. The theory of block intersection polynomials is extended, and new applications of these polynomials to the studies of graphs and block designs are obtained. In particular, we obtain a new method of bounding the size of a clique in an edge-regular graph with given parameters, which can improve on the Hoffman bound when applicable, and a new method for studying the possibility of a graph with given vertex-degree sequence being an induced subgraph of a strongly regular graph with given parameters.     

The connectivity of the block-intersection graphs of designs

It is shown that the vertex connectivity of the block-intersection graph of a balanced incomplete block design,BIBD (v, k, 1), is equal to its minimum degree. A similar statement is proved for the edge connectivity of the block-intersection graph of a pairwise balanced design,PBD (v, K, 1). A partial result on the vertex connectivity of these graphs is also given. Minimal vertex and edge cuts for the corresponding graphs are characterized.Research supported in part by a B.C. Science Council G.R.E.A.T. Scholarship.Research supported in part by an NSERC Postdoctoral Fellowship.     

Ring graphs and complete intersection toric ideals

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideals are minimally generated by Gröbner bases. We prove that any graph can be oriented such that its toric ideal is a complete intersection with a universal Gröbner basis determined by the cycles. It turns out that bipartite ring graphs are exactly the bipartite graphs that have complete intersection toric ideals for any orientation.     

Interval bigraphs are unit grid intersection graphs

We give a short, elementary proof that the interval bigraphs are a strict subclass of the unit grid intersection graphs.     

Symmetric graph designs on friendship graphs

We settle completely the existence of symmetric graph designs on friendship graphs. © 2000 John Wiley & Sons, Inc. J Combin Designs 8:201–206, 2000     

R-domination on block graphs

The k-domination problem is to select a minimum cardinality vertex set D of a graph G such that every vertex of G is within distance k from some vertex of D. We consider a generalization of the k-domination problem, called the R-domination problem. A linear algorithm is presented that solves this problem for block graphs. Our algorithm is a generalization of Slater's algorithm [12], which is applicable for forest graphs.     

Triple intersection numbers for the Paley graphs

We give a tight bound for the triple intersection numbers of Paley graphs. In particular, we show that any three vertices have a common neighbor in Paley graphs of order larger than 25.     

Joint probability generating function for degrees of active/passive random intersection graphs

Correlations of active and passive random intersection graphs are studied in this paper. We present the joint probability generating function for degrees of G ^active(n, m, p) and G ^passive(n, m, p), which are generated by a random bipartite graph G*(n, m, p) on n + m vertices.     

Amply regular graphs and block designs

We study the amply regular diameter d graphs Γ such that for some vertex a the set of vertices at distance d from a is the set of points of a 2-design whose set of blocks consists of the intersections of the neighborhoods of points with the set of vertices at distance d-1 from a. We prove that the subgraph induced by the set of points is a clique, a coclique, or a strongly regular diameter 2 graph. For diameter 3 graphs we establish that this construction is a 2-design for each vertex a if and only if the graph is distance-regular and for each vertex a the subgraph Γ₃(a) is a clique, a coclique, or a strongly regular graph. We obtain the list of admissible parameters for designs and diameter 3 graphs under the assumption that the subgraph induced by the set of points is a Seidel graph. We show that some of the parameters found cannot correspond to distance-regular graphs.     

Edge intersection graphs of linear 3-uniform hypergraphs

Let be the class of edge intersection graphs of linear 3-uniform hypergraphs. It is known that the problem of recognition of the class is NP-complete. We prove that this problem is polynomially solvable in the class of graphs with minimum vertex degree ≥10. It is also proved that the class is characterized by a finite list of forbidden induced subgraphs in the class of graphs with minimum vertex degree ≥16.     

Lower closure and existence theorems for optimal control problems with infinite horizon

Lower closure theorems are proved for optimal control problems governed by ordinary differential equations for which the interval of definition may be unbounded. One theorem assumes that Cesari's property (Q) holds. Two theorems are proved which do not require property (Q), but assume either a generalized Lipschitz condition or a bound on the controls in an appropriateL _p-space. An example shows that these hypotheses can hold without property (Q) holding.     

The shortest distance in random multi‐type intersection graphs

Using an associated branching process as the basis of our approximation, we show that typical inter‐point distances in a multi‐type random intersection graph have a defective distribution, which is well described by a mixture of translated and scaled Gumbel distributions, the missing mass corresponding to the event that the vertices are not in the same component of the graph. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 179–209, 2011     

On the intersection of infinite matroids

We show that the infinite matroid intersection conjecture of Nash-Williams implies the infinite Menger theorem proved by Aharoni and Berger in 2009.We prove that this conjecture is true whenever one matroid is nearly finitary and the second is the dual of a nearly finitary matroid, where the nearly finitary matroids form a superclass of the finitary matroids.In particular, this proves the infinite matroid intersection conjecture for finite-cycle matroids of 2-connected, locally finite graphs with only a finite number of vertex-disjoint rays.     

Simultaneous intersection representation of pairs of graphs

We characterize the pairs (G₁, G₂) of graphs on a shared vertex set that are intersection polysemic: those for which the vertices may be assigned subsets of a universal set such that G₁ is the intersection graph of the subsets and G₂ is the intersection graph of their complements. We also consider several special cases and explore bounds on the size of the universal set. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 171–190, 1999     

Some sufficient conditions for the E- and MV-optimality of block designs having blocks of unequal size

Summary In this paper we consider experimental situations in which ν treatments are to be tested inb blocks whereb _i blocks containk _i experimental units,i=1,...,p, k ₁<k ₂<...<k _p . The idea of a group divisible (GD) design is extended to that of a group divisible design with unequal block sizes (GDUB design) and then a number of results concerning the E- and MV-optimality of GD designs are generalized to the case of GDUB designs.     

Balanced incomplete block designs with block size 8

The necessary conditions for the existence of a balanced incomplete block design on υ ≥ k points, with index λ and block size k, are that: For k = 8, these conditions are known to be sufficient when λ = 1, with 38 possible exceptions, the largest of which is υ = 3,753. For these 38 values of υ, we show (υ, 8, λ ) BIBDs exist whenever λ > 1 for all but five possible values of υ, the largest of which is υ = 1,177, and these five υ's are the only values for which more than one value of λ is open. For λ>1, we show the necessary conditions are sufficient with the definite exception of two further values of υ, and the possible exception of 7 further values of υ, the largest of which is υ=589. In particular, we show the necessary conditions are sufficient for all λ> 5 and for λ = 4 when υ ≠ 22. We also look at (8, λ) GDDs of type 7^m. Our grouplet divisible design construction is also refined, and we construct and exploit α ‐ frames in constructing several other BIBDs. In addition, we give a PBD basis result for {n: n ≡ 0, 1; mod 8, n ≥ 8}, and construct a few new TDs with index > 1. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 233–268, 2001     

A simple closure condition for the normal cone intersection formula

In this paper it is shown that if and are two closed convex subsets of a Banach space and , then whenever the convex cone, , is weak* closed, where and are the support function and the normal cone of the set respectively. This closure condition is shown to be weaker than the standard interior-point-like conditions and the bounded linear regularity condition.

     

The backup 2-median problem on block graphs

The backup 2-median problem is a location problem to locate two facilities at vertices with the minimum expected cost where each facility may fail with a given probability. Once a facility fails, the other one takes full responsibility for the services. Here we assume that the facilities do not fail simultaneously. In this paper, we consider the backup 2-median problem on block graphs where any two edges in one block have the same length and the lengths of edges on different blocks may be different. By constructing a tree-shaped skeleton of a block graph, we devise an O（n log n q- m）-time algorithm to solve this problem where n and m are the number of vertices and edges, respectively, in the given block graph.