A coin tossing game — with a biased coin with probability \(q\) for the tail — for \(n\) persons was discussed by Moritz and Williams in \(1987\), in which the probability for players to go out in a prescribed order is described by what is commonly called the “major index” (due to Major MacMahon), which is an important statistic for the permutation group \(\mathcal{S}_n\). We first describe a variation on this game, for which the same question is answered in terms of the better known statistic “length function” in the sense of Coxeter group theory (also called “inversion number” in combinatorial literature). This entails a new bijection implying the old equality (due to MacMahon) of the generating functions for these two statistics.

Next we describe a game for \(2n\) persons where the ‘same’ question is answered in terms of the Coxeter length function for the reflection group of type \(B_n\). We conclude with some miscellaneous results and questions.

The achromatic index of a graph \(G\) is the largest integer \(k\) admitting a proper colouring of edges of \(G\) in such a way that each pair of colours appears on some pair of adjacent edges. It is shown that the achromatic index of \(K_{12}\) is \(32\).

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.