After introducing and discussing the notion of length two path centered surface area for general graphs, particularly for bipartite graphs, we derive a closed-form expression and an explicit expression for the length two path centered surface areas of the hypercube and the star graph, respectively.

The Ramsey numbers \( r(F, G) \) are investigated, where \( F \) is a non-tree graph of order \( 5 \) and minimum degree \( 1 \), and \( G \) is a connected graph of order \( 6 \). For all pairs \( (F, G) \) where \( F \neq K_5 – K_{1,3} \), the exact value of \( r(F, G) \) is determined. In order to settle \( F = K_5 – K_{1,3} \), we prove \( r(K_5 – K_{1,3}, G) = r(K_4, G) \).

A Stanton-type graph \( S(n, m) \) is a connected multigraph on \( n \) vertices such that for a fixed \( m \) with \( n-1 \leq m \leq \binom{n}{2} \), there is exactly one edge of multiplicity \( i \) (and no others) for each \( i = 1, 2, \ldots, m \). In this note, we show how to decompose \( \lambda K_n \) (for the appropriate minimal values of \( \lambda \)) into Stanton-type graphs \( S(4, 3) \) of the LOE and OLE types.

For a nontrivial connected graph \(G\), an Eulerian walk in \(G\) is a closed walk that contains every edge of \(G\) at least once. An Eulerian walk is irregular if it encounters no two edges of \(G\) the same number of times and the minimum length of an irregular Eulerian walk in \(G\) is the Eulerian irregularity of \(G\). In this work, we determine the Eulerian irregularities of all prisms, grids and powers of cycles.

In this paper, we consider the use of balanced arrays (B-arrays) in constructing discrete fractional factorial designs (FFD) of resolution \((2u+1)\), with \(u=2\) and \(3\), in which each of the \(m\) factors is at two levels (say, \(0\) and \(1\)), denoted by factorial designs of \(2^m\) series. We make use of the well-known fact that such designs can be realized under certain conditions, by using balanced arrays of strength four and six (with two symbols), respectively. Here, we consider the existence of B-arrays of strength \(t=4\) and \(t=6\), and discuss how the results presented can be used to obtain the maximum value of \(m\) for a given set of treatment-combinations. Also, we provide some illustrative examples in which the currently available \(\max(m)\) values have been improved upon.

For integers \( k \geq 1 \), a \((p,q)\)-graph \( G = (V, E) \) is said to admit an AL(\(k\))-traversal if there exists a sequence of vertices \((v_1, v_2, \ldots, v_p)\) such that for each \( i = 1, 2, \ldots, p-1 \), the distance between \( v_i \) and \( v_{i+1} \) is \( k \). We call a graph \( k \)-step Hamiltonian (or say it admits a \( k \)-step Hamiltonian tour) if it has an AL(\(k\))-traversal and \( d(v_1, v_p) = k \). In this paper, we investigate the \( k \)-step Hamiltonicity of graphs. In particular, we show that every graph is an induced subgraph of a \( k \)-step Hamiltonian graph for all \( k \geq 2 \).

A difference system of sets (DSS) is any collection of subsets of \(\mathbb{Z}_n\) with the property that the differences from distinct sets cover \(\mathbb{Z}_n\). That is, every non-zero class in \(\mathbb{Z}_n\) can be written as a difference of classes in at least one way. DSS were introduced by Levenstein in 1971 only for finite fields but the case for just 2 subsets had been previously considered by Clauge. Their work emphasized an application to synchronizable codes. A DSS is triangular if its sets contain only triangular numbers mod \(n\). We show that a triangular DSS cannot exist in \(\mathbb{Z}_{2^k}\) for \(k > 3\).

In testing web applications, details of user visits may be recorded in a web log and converted to test cases. This is called user-session-based testing and studies have shown that such tests may be effective at revealing faults. However, for popular web applications with a larger user base, many user-sessions may build up and test suite management techniques are needed. In this paper, we focus on the problem of test suite prioritization. That is, given a large test suite, reorder the test cases according to a criterion that is hypothesized to increase the rate of fault detection. Previous work shows that 2-way combinatorial-based prioritization is an effective prioritization criterion. We develop a greedy algorithm where we consider memory usage and the time that it takes to prioritize test suites. We represent software tests in a graph by storing unique parameters as vertices and \( n \)-way sets as edges or series of edges. Our experiments demonstrate the efficiency of this approach.

The lattice-simplex covering density problem aims to determine the minimal density by which lattice translates of the \( n \)-simplex cover \( n \)-space. Currently, the problem is completely solved in \( 2 \) dimensions. A computer search on the problem in three dimensions gives experimental evidence that for the simplex \( D \) (the convex hull of the unit basis vectors), the most effective lattice corresponds to the tile known as the \( 84 \)-shape. The \( 84 \)-shape tile has been shown to be a local minimum of the density function. We explain the mechanics behind an algorithm which determines the most efficient lattice in the interior of an arbitrary combinatorial type.

An efficient conditional expectation algorithm for generating covering arrays has established a number of the best known upper bounds on covering array numbers. Despite its theoretical efficiency, the method requires a large amount of storage and time. In order to extend the range of its application, we generalize the method to find covering arrays that are invariant under the action of a group, reducing the search to consider only orbit representatives of interactions to be covered. At the same time, we extend the method to construct a generalization of covering arrays called quilting arrays. The extended conditional expectation algorithm, as expected, provides a technique for generating covering and quilting arrays that reduces the time and storage required. Remarkably, it also improves on the best known bounds on covering array numbers in a variety of parameter situations.