Let \( G = K_{a,b} \), where \( a, b \) are even, or \( G = K_{a,a} – M_{2a} \), where \( a \geq 1 \) is an odd integer and \( M_{2a} \) is a perfect matching in \( K_{a,a} \). It has been shown ([3,4]) that \( G \) is arbitrarily decomposable into closed trails. Billington asked if the graph \( K_{r,s} – F \), where \( s, r \) are odd and \( F \) is a (smallest possible) spanning subgraph of odd degree, is arbitrarily decomposable into closed trails ([2]).

In this article we answer the question in the affirmative.

This paper considers the Lehmer matrix and its recursive analogue. The determinant of the Lehmer matrix is derived explicitly by both its LU and Cholesky factorizations. We further define a generalized Lehmer matrix with \((i,j)\) entries \( g_{ij} = \frac{\text{min} \{u_{i+1}, u_{j+1}\}}{\text{max} \{u_{i+1}, u_{j+1}\}} \) where \( u_n \) is the \( n \)th term of a binary sequence \(\{u_n\}\). We derive both the LU and Cholesky factorizations of this analogous matrix and we precisely compute the determinant.

Random number generators are a small part of any computer simulation project. Yet they are the heart and the engine that drives the project. Often times software houses fail to understand the complexity involved in building a random number generator that will satisfy the project requirements and will be able to produce realistic results. Building a random number generator with a desirable periodicity, that is uniform, that produces all the random permutations with equal probability, and at random, is not an easy task. In this paper we provide tests and metrics for testing random number generators for uniformity and randomness. These tests are in addition to the already existing tests for uniformity and randomness, which we modify by running each test a large number of times on sub-sequences of random numbers, each of length \( n \). The test result obtained each time is used to determine the probability distribution function. This eliminates the random number generator misclassification error. We also provide new tests for uniformity and randomness, the new tests for uniformity test the skewness of each one of the subgroups as well as the kurtosis. The tests for randomness, which include the Fourier spectrum, the phase spectrum, the discrete cosine transform spectrum, and the orthogonal wavelet domain, test for patterns not detected in the raw data space. Finally we provide visual and acoustic tests.

For a connected graph \( G \) of order \( n \), the detour distance \( D(u, v) \) between two vertices \( u \) and \( v \) in \( G \) is the length of a longest \( u-v \) path in \( G \). A Hamiltonian labeling of \( G \) is a function \( c: V(G) \to \mathbb{N} \) such that

\[ |c(u) – c(v)| + D(u,v) \geq n \]

for every two distinct vertices \( u \) and \( v \) of \( G \). The value \( \text{hn}(c) \) of a Hamiltonian labeling \( c \) of \( G \) is the maximum label (functional value) assigned to a vertex of \( G \) by \( c \); while the Hamiltonian labeling number \( \text{hn}(G) \) of \( G \) is the minimum value of a Hamiltonian labeling of \( G \). We present several sharp upper and lower bounds for the Hamiltonian labeling number of a connected graph in terms of its order and other distance parameters.

A graph \( G \) is \( 3 \)-existentially closed (\( 3 \)-e.c.) if each \( 3 \)-set of vertices can be extended in all of the possible eight ways. Results which improve the lower bound of the minimum order of a \( 3 \)-e.c. graph are reported. It has been shown that \( m_{ec}(3) \geq 24 \), where \( m_{ec}(3) \) is defined to be the minimum order of a \( 3 \)-e.c. graph.

In this study, we analyze the structure of the full collineation group of certain Veblen-Wedderburn (VW) planes of orders \( 5^2 \), \( 7^2 \), and \( 11^2 \). We also discuss a reconstruction method using their collineation groups.

A Sarvate-Beam Quad System \( SB(v, 4) \) is a set \( V \) of \( v \) elements and a collection of \( 4 \)-subsets of \( V \) such that each distinct pair of elements in \( V \) occurs \( i \) times for every \( i \) in the list \( 1, 2, \ldots, \binom{v}{2} \). In this paper, we completely enumerate all Sarvate-Beam Quad Systems for \( v = 6 \).

In this paper, we present (by using Cauchy-Schwarz inequalities) some new results amongst the parameters of balanced arrays (B-arrays) with two symbols and having strength four, which are necessary for the existence of such balanced arrays. We then discuss and illustrate their use and applications.

The Oberwolfach problem (OP) asks whether \( K_n \) (for \( n \) odd) or \( K_n \) minus a \( 1 \)-factor (for \( n \) even) admits a \( 2 \)-factorization where each \( 2 \)-factor is isomorphic to a given \( 2 \)-factor \( F \). The order \( n \) and the type of the \( 2 \)-factor \( F \) are the parameters of the problem. For \( n \leq 17 \), the existence of a solution has been resolved for all possible parameters. There are also many special types of \( 2 \)-factors for which solutions to OP are known. We provide solutions to OP for all orders \( n \), \( 18 \leq n \leq 40 \). The computational results for higher orders were obtained using the SHARCNET high-performance computing cluster.