In this paper, we prove that the Equitable \(\Delta\)-Coloring Conjecture holds for planar graphs with maximum degree \(\Delta \geq 13\).

An array \(A[i, j]\), \(1 \leq i \leq n, 1 \leq j \leq n\), has a period \(A[p,p]\) of dimension \(p \times p\) if \(A[i, j] = A[i+p, j+p]\) for \(i, j = 1 \ldots n-p\). The period of \(A_{p,p}\) is the shortest such \(p\).We study two-dimensional pattern matching, and several other related problems, all of which depend on finding the period of an array.In summary, finding the period of an array in parallel using \(p\) processors for general alphabets has the following bounds:
\[
\begin{cases}
\Theta\left(\frac{n^2}{p}\right) & \text{if } p \leq \frac{n^2}{\log \log n}, n>17^3 \quad\quad\quad\quad\quad\quad\quad\quad(1.1) \\
\Theta(\log\log n) & \text{if } \frac{n^2}{\log \log n} < p 17^3 \quad\;\; \quad\quad\quad\quad (1.2) \\
\Theta\left(\log\log_{\frac{2p}{n^2}}{p}\right) & \text{if } n^2 \leq p 17^3 \quad \quad\quad\quad\quad (1.3) \\
\Theta\left(\log\log_{\frac{2p}{n^2}}{p}\right) & \text{if } n^2 \log^2 n \leq p \leq n^4, \text{ $n$ large enough} \quad (1.4)
\end{cases}
\]

In [5] Kløve gave tables of the best bounds known on the size of optimal difference triangle sets. In this note, we give examples of difference triangle sets found by computer search which improve on the upper bounds in [5]. In four cases, these examples are proved to be optimal.

A Latin square \((S, \cdot)\) is said to be \((3, 1, 2)\)-\({conjugate-orthogonal}\) if \(x \cdot y = z \cdot w\), \(x \cdot_{312} y = z \cdot_{312} w\) imply \(x = z\) and \(y = w\), for all \(x, y, z, w \in S\), where \(x_3 \cdot_{312} x_1 = x_2\) if and only if \(x_1 \cdot x_2 = x_3\).Such a Latin square is said to be \({holy}\) \(((3,1,2)\)-HCOLS for short) if it has disjoint and spanning holes corresponding to missing sub-Latin squares.Let \((3,1,2)\)-HCOLS\((h^n)\) denote a \((3,1,2)\)-HCOLS of order \(hn\) with \(n\) holes of equal size \(h\).We show that, for any \(h \geq 1\), a \((3,1,2)\)-HCOLS\((h^n)\) exists if and only if \(n \geq 4\), except \((n,h) = (6,1)\), and except possibly \((n,h) = (10,1)\) and \((4,2t+1)\) for \(t \geq 1\).Let \((3,1,2)\)-ICOILS\((v,k)\) denote an idempotent \((3,1,2)\)-COLS of order \(v\) with a hole of size \(k\).We prove that a \((3,1,2)\)-ICOILS\((v,k)\) exists for all \(v \geq 3k+1\) and \(1 \leq k \leq 5\), except possibly \(k = 4\) and \(v \in \{35, 38\}\).

The main object of this paper is the construction of BIBD’s with \(6 \leq k \leq 11\) and \(\lambda = 1\). These balanced incomplete block designs can be simply constructed from some associated group divisible designs with the number of groups being a prime power, and it is these group divisible designs that are constructed directly. Other related designs are discussed.

We have carried out a large number of computer searches for the base sequences \(BS(n + 1, n)\) as well as for three important subsets known as Turyn sequences, normal sequences, and near-normal sequences. In the Appendix, we give an extensive list of \(BS(n + 1, n)\) for \(n \leq 32\). The existence question for Turyn sequences in \(BS(n + 1, n)\) was resolved previously for all \(n \leq 41\), and we extend this bound to \(n \leq 51\). We also show that the sets \(BS(n + 1, n)\) do not contain any normal sequences if \(n = 27\) or \(n = 28\). To each set \(BS(n + 1, n)\), we associate a finite graph \(\Gamma_{n}\), and determine these graphs completely for \(n \leq 27\). We show that \(BS(m,n) = \emptyset \quad \text{if} \quad m \geq 2n, \; n > 1, \; \text{and} \; m + n \; \text{is odd}\),
and we also investigate the borderline case \(m = 2n – 1\).

In this paper, we look at triangle-free graphs with maximum degree three. By an inequality proved by K. Fraughnaugh\(^*\) in 1990, the number of vertices \(v\), edges \(e\), and independence \(i\) of such a graph satisfy \(e \geq \frac{13v}{2} – 14i\). We prove that there is a unique non-cubic, connected graph for which this inequality is sharp. For the cubic case, we describe a computer algorithm that established that two such extremal cubic graphs exist with \(v = 14\), and none for \(v = 28\) or \(42\). We give a complete list of cubic, and provide some new examples of non-cubic, triangle-free graphs with \(v \leq 36\) and independence ratio \(\frac{i}{v}\) less than \(\frac{3}{8}\).

Fibonacci strings turn out to constitute worst cases for a number of computer algorithms which find generic patterns in strings. Examples of such patterns are repetitions, Abelian squares, and “covers”. In particular, we characterize in this paper the covers of a circular Fibonacci string \(\mathcal{C}(F_k)\) and show that they are \(\Theta(|F_k|^2)\) in number. We show also that, by making use of an appropriate encoding, these covers can be reported in \(\Theta(|F_k|)\) time. By contrast, the fastest known algorithm for computing the covers of an arbitrary circular string of length \(n\) requires time \(O(n \log n)\).

A polychrome labeling of a simple and connected graph \(G = (V, E)\) by an abelian group \(A\) is a bijective map from \(V\) onto \(A\) such that the induced edge labeling \(f^*(vw) = f(v) + f(w)\), \(vw \in E\), is injective. The paper completes the characterization of polychrome paths and cycles begun in [3].

We introduce a generalisation of the concept of a complete mapping of a group, which we call a quasi-complete mapping, and which leads us to a generalised form of orthogonality in Latin squares. In particular, the existence of a quasi-complete mapping of a group is shown to be sufficient for the existence of a pair of Latin squares such that if they are superimposed so as to form an array of unordered pairs, each unordered pair of distinct elements occurs exactly twice. We call such a pair of Latin squares quasi-orthogonal and prove that an abelian group possesses a quasi-complete mapping if and only if it is not of the form \(\mathbb{Z}_{4m+2} \oplus G\), \(|G|\) odd. In developing the theory of quasi-complete mappings, we show that the well-known concept of a quasi-complete Latin square arises quite naturally in this setting. We end the paper by giving a sufficient condition for the existence of a pair of quasi-orthogonal Latin squares which are also quasi-row-complete.