Bollobas posed the problem of finding the least number of edges, \(f(n)\), in a maximally nonhamiltonian graph of order \(n\). Clark, Entringer and Shapiro showed \(f(n) = \left\lceil \frac{3n}{2} \right\rceil\) for all even \(n \geq 36\) and all odd \(n \geq 53\). In this paper, we give the values of \(f(n)\) for all \(n \geq 3\) and show \(f(n) = \left\lceil \frac{3n}{2} \right\rceil\) for all even \(n \geq 20\) and odd \(n \geq 17\).

Three mutually orthogonal idempotent Latin squares of order \(18\) are constructed, which can be used to obtain \(3\) HMOLS of type \(5^{18}\) and type \(23^{18}\) and to obtain a \((90, 5, 1)\)-PMD.

A graph is well-covered if every maximal independent set is also a maximum independent set. A \(1\)-well-covered graph \(G\) has the additional property that \(G – v\) is also well-covered for every point \(v\) in \(G\). Thus, the \(1\)-well-covered graphs form a subclass of the well-covered graphs. We examine triangle-free \(1\)-well-covered graphs. Other than \(C_5\) and \(K_2\), a \(1\)-well-covered graph must contain a triangle or a \(4\)-cycle. Thus, the graphs we consider have girth \(4\). Two constructions are given which yield infinite families of \(1\)-well-covered graphs with girth \(4\). These families contain graphs with arbitrarily large independence number.

A \(d\)-dimensional Perfect Factor is a collection of periodic arrays in which every \(k\)-ary \((n_1, \ldots, n_d)\) matrix appears exactly once (periodically). The one-dimensional case, with a collection of size one, is known as a De Bruijn cycle. The \(1\)- and \(2\)-dimensional versions have proven highly applicable in areas such as coding, communications, and location sensing. Here we focus on results in higher dimensions for factors with each \(n_i = 2\).

It is shown that the existence of a semi-regular automorphism group of order \(m\) of a binary design with \(v\) points implies the existence of an \(n\)-ary design with \(v/m\) points. Several examples are described. Examples of other \(n\)-ary designs are considered which place such \(n\)-ary designs in context among \(n\)-ary designs generally.

Let \(G\) be a connected graph with \(v \geq 3\). Let \(v \in V(G)\). We define \(N_k(v) = \{u|u \in V(G) \text{ and } d(u,v) = k\}\). It is proved that if for each vertex \(v \in V(G)\) and for each independent set \(S \subseteq N_2(v)\), \(|N(S) \cap N(v)| \geq |S| + 1\), then \(G\) is hamiltonian. Several previously known sufficient conditions for hamiltonian graphs follow as corollaries. It is also proved that if for each vertex \(v \in V(G)\) and for each independent set \(S \subseteq N_2(v)\), \(|N(S) \cap N(v)| \geq |S| + 2\), then \(G\) is pancyclic.

We give recursive methods for enumerating the number of orientations of a tree which can be efficiently dominated. We also examine the maximum number, \(\eta_q\), of orientations admitting an efficient dominating set in a tree with \(q\) edges. While we are unable to give either explicit formulas or recursive methods for finding \(\eta_q\), we are able to show that the growth rate of the sequence \(\langle\eta_q\rangle\) stabilizes by showing that \(\lim_{q\to\infty}\eta^\frac{1}{q}_q \) exists.

Let \(G = (V, E)\) be a finite simple graph. \(G\) is said to be a magic graph iff there exists a magic assignment of \(G\), which is a mapping \(L\) from \(E\) to \({N} = \{1, 2, \ldots\}\) such that the sums of the labels of all edges incident to the vertices in \(V\) are identical. Let \(M(G)\) be the set of all magic assignments of \(G\). For any \(L\) in \(M(G)\), define \(s(L) = \max\{L(e): e \in E\}\). Then, the magic strength of \(G\) is defined as \(m(G) = \min\{s(L): L \in M(G)\}\). In this paper, we determine the magic strengths of several classes of graphs and introduce some constructions of magic graphs. We also show that every connected graph is an induced subgraph of a magic graph.

We determine upper bounds on the number of elements in connected and \(3\)-connected matroids with fixed rank and bounded cocircuit size. The existence of these upper bounds is a Ramsey property of matroids. We also determine size type function and extremal matroids in several classes of matroids with small cocircuits.