Orthogonal diagonal latin squares of order n, ODLS(n), are orthogonal latin squares of order n with transversals on both the main diagonal and the back diagonal of each square. It has been proven that ODLS(n) exist for all n except n = 2, 3, 6, 10, 14, 15, 18 and 26, in which the first three are impossible. In this note an example of ODLS(14) is given.

Suppose that in a complete graph on N points, each edge is given arbitrarily either the color red or the color blue, but the total number of blue edges is fixed at T. We find the minimum number of monochromatic triangles in the graph as a function of N and T. The maximum number of monochromatic triangles presents a more difficult problem. Here we propose a reasonable conjecture supported by examples.

A direct construction for partially resolvable t-partitions is presented and then used to give a recursive construction for BIBDs (ν, 4, 2). In particular, we construct BIBD(ν, 4, 2) with BIBD(ν, 4, 2) embedded in it whenever ν = 3u + a, a ∈ {1, 4, 7}. This result allows us to give simple proofs for the existence of BIBD(ν, 4, 2) with various additioinal properties.

We give a constructive proof for the theorem of Lovász and Plummer which asserts the existence of an ear decomposition of a 1-factor covered graph.

S. T. Hedetniemi and P. J. Slater have shown that if G is a triangle-free connected graph with at least three vertices, then

where K(G) is the clique graph of G and K2(G) = K(K(G)) is the first iterated clique graph. In this paper, we generalize the above result to a wider class of graphs.

It has been conjectured that for any union-closed set there exists some element which is contained in at least half the sets in . It is shown that this conjecture is true if the number of sets in is less than 25. Several conditions on a counterexample are also obtained.

An induced subgraph H of connectivity (edge-connectivity) n in a graph G is a major n-connected (major n-edge-connected) subgraph of G if H contains no subgraph with connectivity (edge- connectivity) exceeding n and H has maximum order with respect to this property. An induced subgraph is a major (major edge-) subgraph if it is a major n-connected (major n-edge-connected) subgraph for some n. Let m be the maximum order among all major subgraphs of C. Then the major connectivity set K(G) of G is defined as the set of all n for which there exists a major n-connected subgraph of G having order m. The major edge-connectivity set is defined analogously. The connectivity and the elements of the major connectivity set of a graph are compared, as are the elements of the major connectivity set and the major edge-connectivity set of a graph. It is shown that every set S of nonnegative integers is the major connectivity set of some graph G. Further, it is shown that for each positive integer m exceeding every element of S, there exists a graph G such that every major k-connected subgraph of G, where k ∈ K(G), has order m. Moreover, upper and lower bounds on the order of such graphs G are established.

Regular maps of type {p, q}r and the associated groups Gp,q,r are considered for small values for p, q and r. In particular, it is shown that the groups G4,6,6 and G5,5,6 are Abelian-by-infinite, and there are infinitely many regular maps of each of the types {4, 6}6, {5, 5}6, {5, 6}6 and {6, 6}6.

Let X1, X2, …, Xn be identically distributed independent random variables belonging to the domain of attraction of the normal law, have zero means and Pr{Xr ≠ 0} > 0. Suppose a0, a1, …, an are non-zero real numbers and max and εn is such that as n → ∞, εn. If Nn be the number of real roots of the equation then for n > n0, Nn > εn log n outside an exceptional set of measure at most provided limn→∞ (kn/tn) is finite.

A set with a relation is isomorphic to a group quotient under the condition described as weak homogeneity, and to the quotient of a group with relation preserved by right and left translations if the homogeneity is strengthened. A method of constructing these group quotients and, furthermore, all such very homogeneous spaces, is described and an illustrative example given.

The quintuple product identity has appeared many times in the literature. Indeed, no fewer than 12 proofs have been given. We establish a more general identity from which the quintuple product identity follows in two ways.

A random rooted labelled tree on n vertices has asymptotically the same shape as a branching-type process, in which each generation of a branching process with Poisson family sizes, parameter one, is supplemented by a single additional member added at random to one of the families in that generation. In this note we use this probabilistic representation to deduce the asymptotic distribution of the distance from the root to the nearest endertex other than itself.

Assume G is a graph with m edges. By T(n, G) we denote the classical Turan number, namely, the maximum possible number of edges in a graph H on n vertices without a copy of G. Similarly if G is a family of graphs then H does not have a copy of any member of the family. A Zk-colouring of a graph G is a colouring of the edges of G by Zk, the additive group of integers modulo k, avoiding a copy of a given graph H, for which the sum of the values on its edges is 0 (mod k). By the Zero-Sum Turan number, denoted T(n, G, Zk), k¦m, we mean the maximum number of edges in a Zk-colouring of a graph on n vertices that contains no zero-sum (mod k) copy of G. Here we mainly solve two problems of Bialostocki and Dierker [6].

Problem 1. Determine T(n, tK2, Zk) for ¦|t. In particular, is it true that T(n, tK2, Zk) = T(n, (t+k-1)K2)?

Problem 2. Does there exist a constant c(t, k) such that T(n, Ft, Zk) ≦ c(t, k)n, where Ft is the family of cycles of length at least t?

The Petersen graph on 10 vertices is the smallest example of a vertex-transitive graph which is not a Cayley graph. We consider the problem of determining the orders of such graphs. In this, the first of a series of papers, we present a sequence of constructions which solve the problem for many orders. In particular, such graphs exist for all orders divisible by a fourth power, and all even orders which are divisible by a square.

We characterize all finite linear spaces with p ≤ n2 points where n ≥ 8 for p ≠ n2 − 1 and n ≥ 23 for p = n2−1, and the line range is {n−1, n, n+1}. All such linear spaces are shown to be embeddable in finite projective planes of order a function of n. We also describe the exceptional linear spaces arising from p < n2−1 and n ≥ 4.

A recursive construction for orthogonal diagonal latin squares, using group divisible designs, is presented. In consequence the numbers of orders for which the existence of such squares is in question is reduced to 72.

Let S be a finite linear space on v ≥ n2 –n points and b = n2+n+1–m lines, m ≧ 0, n ≧ 1, such that at most m points are not on n + 1 lines. If m ≧ 1, except if m = 1 and a unique point on n lines is on no line with two points, then S embeds uniquely in a projective plane of order n or is one exceptional case if n =4. If m ≦ 1 and if v ≧ n2 – 2√n + 3, + 6, the same conclusion holds, except possibly for the uniqueness.

1991 Mathematics subject classification (Amer. Math. Soc.) 05 B 05, 51 E 10.

A graph G is divisible by t if its edge set can be partitioned into t subsets, such that the subgraphs (called factors) induced by the subsets are all isomorphic. Such an edge partition is an isomorphic factorization. It is proved that a 2k-regular graph with an even number of vertices is divisble by 2k provided it contains either no 3-cycles or no 5-cycles. It is also shown that any 4-regular graph with an even number of vertices is divisible by 4. In both cases the components of the factors found are paths of length 1 and 2, and the factorizations can be constructed in polynomial time.

A permutation group G on a finite set Ω is always exposable if whenever G stabilises a switching class of graphs on Ω, G fixes a graph in the switching class. Here we consider the problem: given a finite group G, which permutation representations of G are always exposable? We present solutions to the problem for (i) 2-generator abelian groups, (ii) all abelian groups in semiregular representations. (iii) generalised quaternion groups and (iv) some representations of the symmetric group Sn.

Suppose that a graph process begins with n isolated vertices, to which edges are added randomly one by one so that the maximum degree of the induced graph is always at most d. In a previous article, the authors showed that as n → ∞, with probability tending to 1, the result of this process is a d-regular graph. This graph is shown to be connected with probability asymptotic to 1.