Let \(C\) be a perfect 1-error-correcting code of length \(15\). We show that a quotient \(H(C)\) of the minimum distance graph of \(C\) constitutes an invariant for \(C\) more sensible than those studied up to the present, namely the kernel dimension and the rank. As a by-product, we get a nonlinear Vasil’ev code \(C\) all of whose associated Steiner triple systems are linear. Finally, the determination of \(H(C)\) for known families of \(C\)’s is presented.

A computer search shows that there does not exist a nested BIB design \(\text{NB}(10, 15, 2, 3)\).

We construct several families of simple 4-designs, which are closely related to Alltop’s series with parameters \(4-(2^f+1,5,5)\), \(f\) odd. More precisely, for every \(q=2^f\), where \(gcd(f,6)=1\), \(f\geq5\), we construct designs with the following parameters:

\[4-(q+1,6,\lambda),\, \text{where}\, \lambda\in\{60,70,90,100,150,160\},\]
\[4-(q+1,8,35),\]
\[4-(q+1,9,\lambda),\, \text{where}\, \lambda\in\{63,147\}.\]

Eulerian numbers may be defined recursively and have applications to many branches of mathematics. We derive some congruence and divisibility properties of Eulerian numbers.

In this paper, we determine the spectrum of support sizes of indecomposable threefold triple systems of order \(v\) for all \(v > 15\).

Let \(G = (V,E)\) be a graph. A set \(S \subseteq V\) is a dominating set if every vertex not in \(S\) is adjacent to a vertex in \(S\). Furthermore, a set \(S \subseteq V\) is a restrained dominating set if every vertex not in \(S\) is adjacent to a vertex in \(S\) and to a vertex in \(V – S\). The domination number of \(G\), denoted by \(\gamma(G)\), is the minimum cardinality of a dominating set, while the restrained domination number of \(G\), denoted by \(\gamma_r(G)\), is the minimum cardinality of a restrained dominating set of \(G\).
We show that if a connected graph \(G\) of order \(n\) has minimum degree at least \(2\) and is not one of eight exceptional graphs, then \(\gamma_r(G) \leq (n – 1)/2\). We show that if \(G\) is a graph of order \(n\) with \(\delta = \delta(G) \geq 2\), then \(\gamma_r(G) \leq n(1 + (\frac{1}{\delta})^\frac{\delta}{\delta-1} – (\frac{1}{\delta})^\frac{1}{\delta-1})\).

Given a two-dimensional text \(T\) and a set of patterns \(\mathcal{D} = \{P_1, \ldots, P_k\}\) (the dictionary), the two-dimensional \({dictionary\; matching}\) problem is to determine all the occurrences in \(T\) of the patterns \(P_i \in \mathcal{D}\). The two-dimensional \({dictionary\; prefix-matching}\) problem is to determine the longest prefix of any \(P_i \in \mathcal{D}\) that occurs at each position in \(T\). Given an alphabet \(\Sigma\), an \(n \times n\) text \(T\), and a dictionary \(\mathcal{D} = \{P_1, \ldots, P_k\}\), we present an algorithm for solving the two-dimensional dictionary prefix-matching problem. Our algorithm requires \(O(|T| + |\mathcal{D}|(log m + log |\Sigma|))\) units of time, where \(m \times m\) is the size of the largest \(P_i \in \mathcal{D}\). The algorithm presented here runs faster than the Amir and Farach [3] algorithm for the dictionary matching problem by an \(O(log k)\) factor. Furthermore, our algorithm improves the time bound that can be achieved using the Lsuffix tree of Giancarlo [6],[7] by an \(O(k)\) factor.

A connected graph \(G = (V, E)\) is said to be \((a, d)\)-antimagic if there exist positive integers \(a, d\) and a bijection \(g: E \to \{1,2,\ldots,|E|\}\) such that the induced mapping \(f_g: V \to {N}\), defined by \(f_g(v) = \sum\{g(u,v): (u, v) \in E(G)\} \), is injective and \(f_g(V) = \{a,a+d,\ldots,a+(|V|-1)d\}\). We deal with \((a, d)\)-antimagic labelings of the antiprisms.

Let \(s'(G)\) denote the Hall-condition index of a graph \(G\). Hilton and Johnson recently introduced this parameter and proved that \(\Delta(G) \leq s'(G) \leq \Delta(G) + 1\). A graph \(G\) is \(s’\)-Class 1 if \(s'(G) = \Delta(G)\) and is \(s’\)-Class 2 otherwise. A graph \(G\) is \(s’\)-critical if \(G\) is connected, \(s’\)-Class 2, and, for every edge \(e\), \(s'(G – e) < s'(G)\). We use the concept of the fractional chromatic index of a graph to classify \(s’\)-Class 2 in terms of overfull subgraphs, and similarly to classify \(s’\)-critical graphs. We apply these results to show that the following variation of the Overfull Conjecture is true;

A graph \(G\) is \(s’\)-Class 2 if and only if \(G\) contains an overfull subgraph \(H\) with \(\Delta(G) = \Delta(H)\).

We prove that if \(m\) be a positive integer and \(X\) is a totally ordered set, then there exists a function \(\phi: X \to \{1,\ldots,m\}\) such that, for every interval \(I\) in \(X\) and every positive integer \(r \leq |I|\), there exist elements \(x_1 < x_2 < \cdots < x_r\) of \(I\) such that \(\phi(x_{i+1}) \equiv \phi(x_{i}) + 1 \pmod{m}\) for \(i=1,\ldots,r-1\).