Let \(G\) be a simple graph, \(a\) and \(b\) integers and \(f: E(G) \to \{a,a+1,\ldots,b\}\) an integer-valued function with \(\sum_{e\in E(G)} f(e) \equiv 0 \pmod{2}\). We prove the following results:(1) If \(b \geq a \geq 2\), \(G\) is connected and \(\delta(G) \geq \max\left[\frac{b}{2}+2, \frac{(a+b+2)^2}{8a}\right]\), then the line graph \(L(G)\) of \(G\) has an \(f\)-factor;(2) If \(b\geq a \geq 2\), \(G\) is connected and \(\delta(L(G)) \geq \frac{(a+2b+2)^2}{8a}\), then \(L(G)\) has an \(f\)-factor.

We show that a cubic graph is \(\frac{3}{2}\)-tough if and only if it is equal to \(K_4\) or \(K_3 \times K_3\) or else is the inflation of a 3-connected cubic graph.

A directed triple system of order \(v\), denoted \(DT S(v)\), is said to be \(k\)-near-rotational if it admits an automorphism consisting of \(3\) fixed points and \(k\) cycles of length \(\frac{v-3}{3}\). In this paper, we give necessary and sufficient conditions for the existence of \(k\)-near-rotational \(DT S(v)\)s.

A correspondence between decompositions of complete directed graphs with loops into collections of closed trails which partition the edge set of the graph and the variety of all column latin groupoids is established. Subvarieties which consist of column latin groupoids arising from decompositions where only certain trail lengths occur are examined. For all positive integers \(m\), the set of values of \(n\) for which the complete directed graph with loops on a vertex set of cardinality \(n\) can be decomposed in this manner such that all the closed trails have length \(m\) is shown to be the set of all \(n\) for which \(m\) divides \(n^2\).

Let \(X\) be a graph and let \(\alpha(X)\) and \(\alpha'(X)\) denote the domination number and independent domination number of \(X\), respectively. We show that for every triple \((m,k,n)\), \(m \geq 5\), \(2 \leq k \leq m\), \(n > 1\), there exist \(m\)-regular \(k\)-connected graphs \(X\) with \(\alpha'(X) – \alpha(X) > n\). The same also holds for \(m = 4\) and \(k \in \{2,4\}\).

Let \(k\) be an integer greater than one. A set \(S\) of integers is called \(k\)-multiple-free (or \(k\)-MF for short) if \(x \in S\) implies \(kx \notin S\). Let \(f_k(n) = \max\{|A| : A \subseteq [1,n] \text{ is } k\text{-MF}\}\). A subset \(A\) of \([1,n]\) with \(|A| = f_k(n)\) is called a maximal \(k\)-MF subset of \([1,n]\). In this paper, we give a recurrence relation and a formula for \(f_k(n)\). In addition, we give a method for constructing a maximal \(k\)-MF subset of \([1,n]\).

This paper concerns the domination numbers \(\gamma_{k,n}\) for the complete \(k \times n\) grid graphs for \(1 \leq k \leq 10\) and \(n \geq 1\). These numbers were previously established for \(1 \leq k \leq 4\). Here we present dominating sets for \(5 \leq k \leq 10\) and \(n \geq 1\). This gives upper bounds for \(\gamma_{k,n}\) for \(k\) in this range. We discuss evidence that indicates that these upper bounds are also lower bounds.

By a graph we mean an undirected simple graph. The genus \(\gamma(G)\) of a graph \(G\) is the minimum genus of the orientable surface on which \(G\) is embeddable. The thickness \(\Theta(G)\) of \(G\) is the minimum number of planar subgraphs whose union is \(G\).

In [1], it is proved that, if \(\gamma(G) = 1\), then \(\Theta(G) = 2\). If \(\gamma(G) = 2\), the known best upper bound on \(\Theta(G)\) is \(4\) and, as far as the author knows, the known best lower bound is \(2\). In this paper, we prove that, if \(\gamma(G) = 2\), then \(\Theta(G) \leq 3\).

A generalization of (binary) balanced incomplete block designs is to allow a treatment to occur in a block more than once, that is, instead of having blocks of the design as sets, allow multisets as blocks. Such a generalization is referred to as an \(n\)-ary design. There are at least three such generalizations studied in the literature. The present note studies the relationship between these three definitions. We also give some results for the special case when \(n\) is \(3\).

We investigate searching strategies for the set \(\{1, \ldots, n\}\) assuming a fixed bound on the number of erroneous answers and forbidding repetition of questions. This setting models the situation when different processors provide answers to different tests and at most \(k\) processors are faulty. We show for what values of \(k\) the search is feasible and provide optimal testing strategies if at most one unit is faulty.