Let Γ be a graph with isomorphic subgraphs G and H, and let θ: G → H be an isomorphism. If θ can be extended to an automorphism of Γ, we call θ a partial automorphism of Γ.

We consider the application of partial automorphisms to the graph reconstruction conjecture, in particular, to the problem of reconstructing graphs with two vertices of degree k – 1 and the remaining vertices of degree k.

Recently, we have introduced the notion of stable permutations in a Latin rectangle L(r×c) of r rows and c columns. In this note, we prove that the set of all stable permutations in L (r×c) forms a distributive lattice which is Boolean if and only if c ≤ 2.

A construction for balanced ternary designs is given. Based on the designs so obtained, a construction of partially balanced ternary designs is given, which gives balanced ternary designs and series of symmetric balanced ternary designs in special cases.

Two graphs, the edge crossing graph E and the triangle graph T are associated with a simple lattice polygon. The maximal independent sets of vertices of E and T are derived including a formula for the size of the fundamental triangles. Properties of E and T are derived including a formula for the size of the maximal independent sets in E and T. It is shown that T is a factor graph of edge-disjoint 4-cycles, which gives corresponding geometric information, and is a partition graph as recently defined by the authors and F. Harary.

A balanced directed cycle design with parameters (υ, k, 1), sometimes called a (υ, k, 1)-design, is a decomposition of the complete directed graph into edge disjoint directed cycles of length k. A complete classification is given of (υ, k, 1)-designs admitting the holomorph {øa, b: x ↦ ax + b∣ a, b ∈ Zυ, (a, υ1) = 1} of the cyclic group Zυ as a group of automorphisms. In particular it is shown that such a design exists if and ony if one of (a) k = 2, (b) p ≡ 1 (mod k) for each prime p dividing υ, or (c) k is the least prime dividing υ, k2 does not divide υ, and p ≡ 1 (mod k) for each prime p < k dividing υ.

Let Gbe a primitive permutation group on a finite set Ω. We investigate the subconstitutents of G, that is the permutation groups induced by a point stabilizer on its orbits in Ω, in the cases where Ghas a diagonal action or a product action on Ω. In particular we show in these cases that no subconstituent is doubly transitive. Thus if G has a doubly transitive subconstituent we show that G has a unique minimal normal subgroup N and either N is a nonabelian simple group or N acts regularly on Ω: we investigate further the case where N is regular on Ω.

We discuss the problem of constructing large graceful trees from smaller ones and provide a partial answer in the case of the product tree Sm {g} by way of a sample of sufficient conditions on g. Interlaced trees play an important role as building blocks in our constructions, although the resulting valuations are not always interlaced.

A Bhaskar Rao design is obtained from the incidence matrix of a partially balanced incomplete block design with m associate classes by negating some elements of the matrix in such a way that the inner product of rows α and β is ci if α and β are ith associates. In this paper we use nested designs constructed from unions of cyclotomic classes to give Bhaskar Rao designs.

A directed packing of pairs into quadruples is a collection of 4-subsets of a set of cardinality ν with the property that each ordered pair of elements appears at most once in a 4-subset (or block). The maximal number of blocks with this property is denoted by DD(2, 4, ν). Such a directed packing may also be thought of as a packing of transtivie tournaments into the complete directed graph on ν points. It is shown that, for all but a finite number of values of ν, DD(2, 4, ν) is maximal.

Cubic Moore graphs of diameter k on 3.2k−2 vertices do not exist for k > 2. This paper exhibits the first known case of nonexistence for generalized cubic Moore graphs when the number of vertices is just less than the critical number for a Moore graph: the generalized Moore graph on 44 vertices does not exist.

Frames have been defined as a certain type of generalization of Room square. Frames have proven useful in the construction of Room squares, in particular, skew Room squares.

We generalize the definition of frame and consider the construction of Room squares and skew Room squares using these more general frames.

We are able to construct skew Room squares of three previously unknown sides, namely 93, 159, and 237. This reduces the number of unknown sides to four: 69, 87, 95 and 123. Also, using this construction, we are able to give a short proof of the existence of all skew Room squares of (odd) sides exceeding 123.

Finally, this frame construction is useful for constructing Room squares with subsquares. We can also construct Room squares “missing” subsquares of sides 3 and 5. The “missing” subsquares of sides 3 and 5 do not exist, so these incomplete Room squares cannot be completed to Room squares.

One way of constructing a 2 – (11,5,4) design is to take together all the blocks of two 2 – (11,5,2) designs having no blocks in common. We show that 58 non-isomorphic 2 – (11,5,4) designs can be so made and that through extensions by complementation these can be packaged into just 12 non-isomorphic reducible 3 – (12,6,4) designs.

An (n + 1, n2 + n + 1)-packing is a collection of blocks, each of size n + 1, chosen from a set of size n2 + n + 1, such that no pair of points is contained in more than one block. If any two blocks contain a common point, then the packing can be extended to a projective plane of order n, provided the number of blocks is sufficiently large. We study packings which have a pair of disjoint blocks (such a packing clearly cannot be extended to a projective plane of order n). No such packing can contain more than n2 + n/2 blocks. Also, if n is the order of a projective plane, then we can construct such a packing with n2 + 1 blocks.

This paper inverstigates the automorphism groups of Cayley graphs of metracyclic p-gorups. A characterization is given of the automorphism groups of Cayley grahs of a metacyclic p-group for odd prime p. In particular, a complete determiniation of the automophism group of a connected Cayley graph with valency less than 2p of a nonabelian metacyclic p-group is obtained as a consequence. In subsequent work, the result of this paper has been applied to solve several problems in graph theory.

A series of balanced incomplete block (BIB) designs with parameters , is constructed, where , w any integer.

Let 〈fn≧0 be nonnegative real numbers with generating function f(x) = Σfnxn. Assume f(x) has the following properties: it has a finite nonzero radius of convergence x0 with its only singularity on the circle of convergence at x = x0 and f(x0) converges to y0; y = f(x) satisfies an analytic identity F(x, y) = 0 near (x0, y0); Fy(l) (x0, y0)= 0, 0 ≦ i < k and Fy(k) (x0, y0) ≠ 0. There are constants γ, a positive rational, and c such that fn~cx0−n n−(1 +ggr;). Furthermore, we show (i) in all cases how to determine γ and c from f(x) and (ii) in certain cases how to determine them from F(x, y).

A balanced tournament design, BTD(n), defined on a 2n—set V is an arrangement of the () distinct unordered pairs of the elements of V into an n × 2n − 1 array such that (1) every element of V is contained in precisely one cell of each column, and (2) every element of V is contained in at most two cells of each row. In this paper, we investigate the existence of balanced tournament designs with a pair of almost orthogonal resolutions. These designs can be used to construct doubly resolvable (ν, 3, 2)- BIBD s and, in our smallest applications, have been used to construct previously unknown doubly resolvable (ν, 3, 2)- B I B D s.

It is well-known that if G is a multigraph (that is, a graph with multiple edges), the maximum number of pairwise disjoint edges in G is ν(G) and its maximum degree is D(G), then |E(G)| ≤ ν [3D/2’. We extend this theorem for r-graphs (that is, families of r-element sets) and for r-multihypergraphs (that is, r-graphs with repeated edges). Several problems remain open.

Let g(n, m) denote the maximal number of distinct rows in any (0, 1 )-matrix with n columns, rank < n, – 1, and all row sums equal to m. This paper determines g(n, m) in all cases:

In addition, it is shown that if V is a k-dimensional vector subspace of any vector space, then V contains at most 2k vectors all of whose coordinates are 0 or 1.

We show that for all n ≥ 3k + 1, n ≠ 6, there exists an incomplete self-orthogonal latin square of order n with an empty order k subarray, called an ISOLS(n;k), except perhaps when (n;k) ∈ {(6m + i;2m):i = 2, 6}.