A labeling of the vertices of a graph with distinct natural numbers induces a natural labeling of its edges: the label of an edge \((x,y)\) is the absolute value of the difference of the labels of \(x\) and \(y\). We say that a labeling of the vertices of a graph of order \(n\) is minimally \(k\)-equitable if the vertices are labeled with \(1, 2, \dots, n\) and in the induced labeling of its edges, every label either occurs exactly \(k\) times or does not occur at all. In this paper, we prove that Butterfly and Benes networks are minimally \(2^r\)-equitable where \(r\) is the dimension of the networks.

The method of large scale group testing has been used in the economical testing of blood samples, and in non-testing situations such as experimental designs and coding theory, for over 50 years. Some very basic questions addressing the minimum number of tests required to identify defective samples still remain unsolved, including the situation where one defective sample in each of two batches are to be found. This gives rise to an intriguing graph theoretical conjecture concerning bipartite graphs, a conjecture which in this paper is proved to be true in the case where vertices in one part of the bipartite graph have low degree.

Using the action of the linear fractional groups \( L_2(q) \), where \( q = 8, 25, 27, 29, 31 \) and \( 32 \), some \( 1 \)-designs are constructed. It is shown that subgroups of the automorphism group of \( L_2(q) \) appear as the full automorphism group of the constructed designs. In the cases \( q = 8 \) and \( 32 \), it is shown that the symmetric groups \( \mathbb{S}_9 \) and \( \mathbb{S}_{33} \), respectively, appear as the automorphism group of one of the constructed designs.

Here we consider string matching problems that arise naturally in applications to music retrieval. The \( \delta \)-Matching problem calculates, for a given text \( T_{1..n} \) and a pattern \( P_{1..m} \) on an alphabet of integers, the list of all indices \( \mathcal{I}_\delta = \{1 \leq i \leq n-m+1 : \max_{j=1}^m \{|P_j – T_{i+j-1}| \leq \delta\}\} \). The \( \gamma \)-Matching problem computes, for given \( T \) and \( P \), the list of all indices \( \mathcal{I}_\gamma = \{1 \leq i \leq n-m+1 : \sum_{j=1}^m |P_j – T_{i+j-1}| \leq \gamma\} \). In this paper, we extend the current result on the different matching problems to handle the presence of “\emph{don’t care}” symbols. We present efficient algorithms that calculate \( \mathcal{I}_\delta \), \( \mathcal{I}_\gamma \), and \( \mathcal{I}_{(\delta,\gamma)} = \mathcal{I}_\delta \cap \mathcal{I}_\gamma \) for pattern \( P \) with occurrences of “don’t cares”.

In 2004, Kim and Nakprasit showed that the chromatic number of \( K_2(9,4) \) is at least 11. In this note we present an 11-coloring for \( K^2(9,4) \) (the square of the Kneser graph \( K(9,4) \)). This proves that the chromatic number of \( K^2(9,4) \) is 11.

Let \( N \) and \( Z \) denote respectively the set of all nonnegative integers and the set of all integers. A \((p,q)\)-graph \( G = (V, E) \) is said to be additively \((a,r)\)-geometric if there exists an injective function \( f : V \to Z \) such that \( f^+(E) = \{a, ar, \dots, ar^{q-1}\} \) where \( a, r \in N \), \( r > 1 \), and \( f^+ \) is defined by \( f^+(uv) = f(u) + f(v) \) for all \( uv \in E \). If further \( f(v) \in N \) for all \( v \in V \), then \( G \) is said to be additively \((a,r)^*\)-geometric. In this paper we characterise graphs which are additively geometric and additively \(^*\)-geometric.

In this paper we find ten new weighing matrices of order \(2n\) and weight \(2n – 5\) constructed from two circulants, by forming a conjecture on the locations of the five zeros in a potential solution. Establishing patterns for the locations of zeros in sequences that can be used to construct weighing matrices seems to be a worthwhile path to explore, as it reduces significantly the computational complexity of the problem.

A dominating set \( S \) in a graph \( G \) is said to be perfect if every vertex of \( G \) not in \( S \) is adjacent to just one vertex of \( S \). Given a vertex subset \( S’ \) of a side \( P_m \) of an \( m \times n \) grid graph \( G \), the perfect dominating sets \( S \) in \( G \) with \( S’ = S \cap V(P_m) \) can be determined via an exhaustive algorithm \( \ominus \) of running time \( O(2^{m+n}) \). Extending \( \ominus \) to infinite grid graphs of width \( m – 1 \), periodicity makes the binary decision tree of \( \ominus \) prunable into a finite threaded tree, a closed walk of which yields all such sets \( S \). The graphs induced by the complements of such sets \( S \) can be codified by arrays of ordered pairs of positive integers via \( \ominus \), for the growth and determination of which a speedier algorithm exists. A recent characterization of grid graphs having total perfect codes \( S \) (with just \( 1 \)-cubes as induced components), due to Klostermeyer and Goldwasser, is given in terms of \( \ominus \), which allows to show that these sets \( S \) are restrictions of only one total perfect code \( S_1 \) in the integer lattice graph \( \Lambda \) of \( \textbf{R}^2 \). Moreover, the complement \( \Lambda – S_1 \) yields an aperiodic tiling, like the Penrose tiling. In contrast, the parallel, horizontal, total perfect codes in \( \Lambda \) are in \( 1-1 \) correspondence with the doubly infinite \( \{0, 1\} \)-sequences.

Let \( G \) be a graph of size \( n \) with vertex set \( V(G) \) and edge set \( E(G) \). A \( \rho \)-\emph{labeling} of \( G \) is a one-to-one function \( f : V(G) \to \{0,1,\dots,2n\} \) such that \( \{|f(u) – f(v)| : \{u,v\} \in E(G)\} = \{x_1,x_2,\dots,x_n\} \), where for each \( i \in \{1,2,\dots,n\} \) either \( x_i = i \) or \( x_i = 2n+1-i \). Such a labeling of \( G \) yields a cyclic \( G \)-decomposition of \( K_{2n+1} \). It is conjectured by El-Zanati and Vanden Eynden that every 2-regular graph \( G \) admits a \( \rho \)-labeling. We show that the union of up to ten vertex-disjoint \( C_{4x+1} \) admits a \( \rho \)-labeling.

The Hamilton-Waterloo problem in the case of triangle-factors and Hamilton cycles asks for a \(2\)-factorization of \( K_n \), in which each \(2\)-factor is either a Hamilton cycle or a triangle-factor. Necessarily \( n \equiv 3 \pmod{6} \). The case of \( n \equiv 9 \pmod{18} \) was completely solved in 2004 by Horak, Nedela, and Rosa. In this note, we solve the problem when \( n \equiv 3 \pmod{18} \) and there are at least two Hamilton cycles. A companion paper treats the case when there is exactly one Hamilton cycle and \( n \equiv 3 \pmod{6} \).

There exist \( 3 \) near bowtie systems of order \( 7 \), \( 12 \) bowtie systems of order \( 9 \), and \( 1{,}411{,}422 \) balanced bowtie systems of order \( 13 \).

Chessboard separation problems are modifications to classic chessboard problems, such as the \( N \) Queens Problem, in which obstacles are placed on the chessboard. This paper focuses on a variation known as the \( N + k \) Queens Problem, in which \( k \) Pawns and \( N + k \) mutually non-attacking Queens are to be placed on an \( N \)-by-\( N \) chessboard. Results are presented from performance studies examining the efficiency of sequential and parallel programs that count the number of solutions to the \( N + k \) Queens Problem using traditional backtracking and dancing links. The use of Stochastic Local Search for determining the existence of solutions is also presented. In addition, preliminary results are given for a similar problem, the \( N + k \) Amazons.

In this paper, it is shown that the graph obtained by overlapping the cycle \( C_m \) (\( m \geq 3 \)) and the complete bipartite graph \( K_{3,3} \) at an edge is uniquely determined by its chromatic polynomial.

A graph \( G \) is said to be in the collection \( M_t \) if there are precisely \( t \) different sizes of maximal independent sets of vertices in \( G \). For \( G \in M_t \), and \( v \in G \), we determine the extreme values that \( x \) can assume where \( G \setminus \{v\} \) belongs to \( M_x \). For both the minimum and maximum values, graphs are given that achieve them, showing that the bounds are sharp. The effect of deleting an edge from \( G \) on the number of sizes of maximal independent sets is also considered.

The chromatic polynomial of a graph \( G \), \( P(G; \lambda) \), is the polynomial in \( \lambda \) which counts the number of distinct proper vertex \( \lambda \)-colorings of \( G \), given \( \lambda \) colors. We compute \( P(C_4 \times P_n; \lambda) \) and \( P(C_5 \times P_n; \lambda) \) in matrix form and will find the generating function for each of these sequences.

The \( n \)-cube is the graph whose vertices are all binary words of length \( n > 1 \) and whose edges join vertices that differ in exactly one entry; i.e., are at Hamming distance \( 1 \) from each other. If a word has a non-empty prefix, not the entire word, which is also a suffix, then it is said to be bordered. A word that is not bordered is unbordered. Unbordered words have been studied extensively and have applications in synchronizable coding and pattern matching. The neighborhood of an unbordered word \( w \) is the word itself together with the set of words at Hamming distance \( 1 \) from \( w \). Over the binary alphabet, the neighborhood of an unbordered word \( w \) always contains two bordered words obtained by complementing the first and last entries of \( w \). We determine those unbordered words \( w \) whose neighborhoods otherwise contain only unbordered words.

Let \( G \) be a graph with vertex set \( V \) and edge set \( E \). A labeling \( f : V \to \{0,1\} \) induces a partial edge labeling \( f^* : E \to \{0,1\} \) defined by \( f^*(xy) = f(x) \) if and only if \( f(x) = f(y) \) for each edge \( xy \in E \). The balance index set of \( G \), denoted \( \text{BI}(G) \), is defined as \( \{|f^{*-1}(0) – f^{*-1}(1)| : |f^{-1}(0) – f^{-1}(1)| \leq 1\} \). In this paper, we study the balance index sets of graphs which are \( L \)-products with cycles and complete graphs.

For two vertices \( u \) and \( v \) in a connected graph \( G \), the detour distance \( D(u,v) \) between \( u \) and \( v \) is the length of a longest \( u – v \) path in \( G \). The detour diameter \( \text{diam}_D(G) \) of \( G \) is the greatest detour distance between two vertices of \( G \). Two vertices \( u \) and \( v \) are detour antipodal in \( G \) if \( D(u,v) = \text{diam}_D(G) \). The detour antipodal graph \( \text{DA}(G) \) of a connected graph \( G \) has the same vertex set as \( G \) and two vertices \( u \) and \( v \) are adjacent in \( \text{DA}(G) \) if \( u \) and \( v \) are detour antipodal vertices of \( G \). For a connected graph \( G \) and a nonnegative integer \( r \), define \( \text{DA}^r(G) \) as \( G \) if \( r = 0 \) and as the detour antipodal graph of \( \text{DA}^{r-1}(G) \) if \( r > 0 \) and \( \text{DA}^{r-1}(G) \) is connected. Then \( \{\text{DA}^r(G)\} \) is the detour antipodal sequence of \( G \). A graph \( H \) is the limit of \( \{\text{DA}^r(G)\} \) if there exists a positive integer \( N \) such that \( \text{DA}^r(G) \cong H \) for all \( r \geq N \). It is shown that \( \{\text{DA}^r(G)\} \) converges if \( G \) is Hamiltonian. All graphs that are the limit of the detour antipodal sequence of some Hamiltonian graph are determined.

For a vertex \( x \) in a graph \( G \), we define \( \Psi_1(x) \) to be the number of edges in the closed neighborhood of \( x \). Vertex \( x^* \) is a neighborhood champion if \( \Psi_1(x^*) > \Psi_1(x) \) for all \( x \neq x^* \). We also refer to such an \( x^* \) as a unique champion. For \( d \geq 4 \), let \( n_0(1,d) \) be the smallest number such that for every \( n \geq n_0(1,d) \) there exists an \( n \)-vertex \( d \)-regular graph with a unique champion. Our main result is that \( n_0(1,d) \) satisfies \( d+3 \leq n_0(1,d) < 3d+1 \). We also observe that there can be no unique champion vertex when \( d = 3 \).

In this paper, we consider the non-existence of some bi-level orthogonal arrays (O-arrays) of strength six, with \( m \) constraints (\( 6 \leq m \leq 32 \)), and with index set \( \mu \) (\( 1 \leq \mu \leq 512 \)). The results presented here tend to improve upon the results available in the literature.

We present constructions and results about GDDs with two groups and block size five in which each block has configuration \((s, t)\), that is, in which each block has exactly \(s\) points from one of the two groups and \(t\) points from the other. After some results for a general \(k\), \(s\), and \(t\), we consider the \((2,3)\) case for block size \(5\). We give new necessary conditions for this family of GDDs and give minimal or near-minimal index examples for all group sizes \(n \geq 4\) except for \(n = 24s + 17\).

The covering number for a subset of leaves in a finite rooted tree is defined as the number of subtrees that remain after deleting all the paths connecting the root to the other leaves. We find the formula for the total sum (hence the average) of the covering numbers for a given subset of labeled leaves over all unordered binary trees with \( n \) leaves.