A disjoint multiple paths problem asks if there exist paths between a given set of vertices. Constraints are applied so that paths are not allowed to share vertices (vertex disjoint multiple paths) or share edges (edge disjoint multiple paths). The vertex disjoint multiple paths problem is one of the classic NP-complete problems presented by Karp [1]. The edge disjoint multiple paths problem is also NP-complete since it is easily transformed from the vertex disjoint multiple paths problem. Because of its importance in electronic circuit design, studies are done for restricted cases. The edge disjoint multiple paths problem remains NP-complete for acyclic graphs and planar graphs. Furthermore, the edge disjoint multiple paths problem remains NP-complete if the graph is limited to an undirected mesh.

In this paper, the edge disjoint multiple paths problem when constructed over a directed mesh is discussed. We found that the multiple paths problem remains NP-complete in this special case. Three polynomial time algorithms are presented in which the following restrictions are made: (i) disjoint paths with the same origin row, the same destination row, distinct origin columns, and distinct destination columns, (ii) disjoint paths with the same origin column, the same destination column, distinct origin rows, and distinct destination rows, and (iii) disjoint paths with the same origin row, distinct origin columns, and distinct destination rows.

We discuss a transform on the set of rational functions over the finite field \( \mathbb{F}_q \). For a subclass of these functions, the transform yields a polynomial and its factorization as a product of the set of monic irreducible polynomials, all of which share a common property \( P \) that depends on the choice of rational function. A general formula is derived from the factorization for the number of monic irreducible polynomials of degree \( n \) having property \( P \). However, it is also possible in some instances to exploit the properties of the factorization to obtain a “closed” form of the answer more directly. We illustrate the method with four examples, two of which appear in the literature. In particular, we give alternative proofs for a result of L. Carlitz on the number of monic irreducible self-reciprocal polynomials and a remarkable result of S. D. Cohen on the number of \((r, m)\)-polynomials, that is, monic irreducible polynomials of the form \( f(x^r) \) of degree \( mr \). We also give a generalization of the factorization of \( x^{q-1} – 1 \) over \( \mathbb{F}_q \) that includes the factorization of \( x^{(q-1)^2} – 1 \). The new results concern translation invariant polynomials, which lead to a consideration of the orders of elements in \( \overline{\mathbb{F}}_q \), the algebraic closure of \( \mathbb{F}_q \). We show that there are an infinite number of \( \theta \in \overline{\mathbb{F}}_q \) such that \( \text{ord}(\theta) \) and \( \text{ord}(r(\theta)) \) are related, in the sense that given one, one can infer information about the other.

Let \( G \) be a graph with \( v \) vertices. If there exists a collection of lists of colors \(\{S_1, S_2, \ldots, S_v\}\) on its vertices, each of size \( k \), such that there exists a unique proper coloring for \( G \) from this list of colors, then \( G \) is called a \({uniquely \;k \;-list\; colorable \;graph}\). In this note, we present a uniquely \( 3 \)-list colorable, planar, and \( K_4 \)-free graph. It is a counterexample to a conjecture by Ch. Eslahchi, M. Ghebleh, and H. Hajiabolhassan [3].

For any integers \( k, d \geq 1 \), a \((p, q)\)-graph \( G \) with vertex set \( V(G) \) and edge set \( E(G) \), where \( p = |V(G)| \) and \( q = |E(G)| \), is said to be \((k, d)\)-strongly indexable (in short \((\textbf{k, d})\)-\textbf{SI}) if there exists a pair of functions \((f, f^+)\) that assigns integer labels to the vertices and edges, i.e., \( f: V(G) \to \{0, 1, \dots, p-1\} \) and \( f^+: E(G) \to \{k, k+d, k+2d, \dots, k+(q-1)d\} \), such that \( f^+(u, v) = f(u) + f(v) \) for any \((u, v) \in E(G)\). We determine here classes of spiders that are \((1, 2)\)-SI graphs. We show that every given \((1, 2)\)-SI spider can be extended to an \((1, 2)\)-SI spider with arbitrarily many legs.

In this paper, we obtain some new results, using inequalities such as Hölder and Minkowski, etc., on the existence of balanced arrays (B-arrays) with two levels and of strength six. We then discuss the use of these results to obtain the maximum number of constraints for B-arrays with given values of the parameter vector \(\underline{\mu}’\). We also include some illustrative examples.

A construction of a minimum cycle basis for the wreath product of a star by a path, two stars and a star by a wheel is given. Moreover, the basis numbers of these products are determined.

For any \( h \in \mathbb{N} \), a graph \( G = (V, E) \) is said to be \( h \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_h \setminus \{0\} \) such that the induced vertex labeling \( l^+: V(G) \to \mathbb{Z}_h \), defined by

\[ l^+(v) = \sum_{uv \in E(G)} l(uv), \]

is a constant map. When this constant is \( 0 \), we call \( G \) a zero-sum \( h \)-magic graph. The null set of \( G \) is the set of all natural numbers \( h \in \mathbb{N} \) for which \( G \) admits a zero-sum \( h \)-magic labeling. A graph \( G \) is said to be uniformly null if every magic labeling of \( G \) induces a zero sum. In this paper, we will identify the null sets of certain planar graphs such as wheels and fans.

In this paper, we find six new weighing matrices of order \( 2n \) and weight \( 2n-9 \) constructed from two circulants, by establishing various patterns on the locations of the nine zeros in a potential solution.

In the past few years, several studies have appeared that relate to the existence of \( \mathbb{Z} \)-cyclic directed-triplewhist tournaments and \( \mathbb{Z} \)-cyclic ordered-triplewhist tournaments. In these studies, the number of players in the tournament is taken to be a prime \( p \) of the form \( p \equiv 2^k + 1 \pmod{2^k+1} \), where \( k \geq 2 \). For the cases \( k = 2, 3, 4 \) it has been shown [6,4,5,12] that \( \mathbb{Z} \)-cyclic directed-triplewhist tournaments and \( \mathbb{Z} \)-cyclic ordered-triplewhist tournaments exist for all such primes except for the impossible cases \( p = 5, 13, 17 \). For the cases \( k = 5, 6, 7 \) it has been shown [13] that \( \mathbb{Z} \)-cyclic directed-triplewhist tournaments exist for all such primes less than \( 3{,}200{,}000 \) and that \( \mathbb{Z} \)-cyclic ordered-triplewhist tournaments exist for all such primes less than \( 3{,}200{,}000 \) with the exception that existence or non-existence of these designs for \( p = 97, 193, 449, 577, 641, 1409 \) is an open question. Here the case \( k = 8 \) is considered. It is established that \( \mathbb{Z} \)-cyclic directed-triplewhist tournaments and \( \mathbb{Z} \)-cyclic ordered-triplewhist tournaments exist for all primes \( p \equiv 257 \pmod{512} \), \( p \leq 6{,}944{,}177 \), except possibly for \( p = 257, 769, 3329 \). For \( p = 3329 \) we are able to construct a \( \mathbb{Z} \)-cyclic directed-triplewhist tournament, but the existence of a \( \mathbb{Z} \)-cyclic ordered-triplewhist tournament remains an open question. Furthermore, for each type of design it is conjectured that our basic constructions will produce these designs whenever \( p > 5{,}299{,}457 \).

The domination graph of a digraph \( D \), denoted \( \text{dom}(D) \), is created using the vertex set of \( D \) and edge \( uv \in E(\text{dom}(D)) \) whenever \( (u,z) \in A(D) \) or \( (v,z) \in A(D) \) for any other vertex \( z \in V(D) \). Specifically, we consider directed graphs whose underlying graphs are isomorphic to their domination graphs. In particular, digraphs are completely characterized where \( UG^c(D) \) is the union of two disjoint paths.