We derive a first-order recurrence for \(a_n(t) = \sum_{k=0}^{n} \frac{(-1)^{n-k}}{1+tk} \binom{n}{k}\) (\(t\) fixed \(t\neq -\frac{1}{m}, m\in \mathbb{N}\)). The first-order recurrence yields an alternative proof for Riordan’s theorem: \(a_n(t) = \binom{1/{t+n}}{n}^{-1}(-1)^n\) and also yields the ordinary generating function \(\sum_{n=0}^{\infty} a_n(t) x^n\) for \(t \in \mathbb{N}.\)From the latter, one easily computes \(\sum_{n=0}^{\infty}a_n(t)\) which turns out to be the well-known \(\sum_{n=0}^{\infty} \frac{(-1)^n}{n+1} = \ln 2\) for \(t=1\). For \(t=2\), we get \(\sum_{n=0}^{\infty} (-1)^n\frac{2n}{(2n+1)} = \frac{\ln(\sqrt{2}+1)}{\sqrt{2}}\).

Avis has shown that the number of vertices of a minimal triangle-free \(5\)-chromatic graph is no fewer than \(19\). Mycielski has shown that this number is no more than \(23\). In this paper, we improve these bounds to \(21\) and \(22\), respectively.

By a refinement of a rank argument used to prove a directed version of the Graham-Pollak theorem, we show that \(n\) bicliques are needed to partition the arc-set of the complement of a directed cycle.

In this paper, we obtain a polynomial inequality of degree three in \(m\) (the number of constraints), with coefficients involving the parameters \(\mu_i\)’s, on the existence of balanced arrays of strength four and with two symbols. Applications of the inequality to specific balanced arrays for obtaining an upper bound on the number of constraints are also discussed.

Let \(r(G)\) denote the rank, over the field of rational numbers, of the adjacency matrix of a graph \(G\). Van Nuffelen and Ellingham have obtained several inequalities which relate \(r(G)\) to other graph parameters such as chromatic number, clique number, Dilworth number, and domination number. We obtain additional results of this type. Our main theorem is that for graphs \(G\) having no isolated vertices, \(OIR(G) \leq r(G)\), where \(OIR(G)\) denotes the upper open irredundance number of \(G\).

Let \(D\) denote any balanced ternary design with block size three, index two, and \(\rho_2 = 1\) (that is, with each element occurring repeated in just one block). This paper shows that there exists such a design \(D\) on \(V\) elements containing exactly \(k\) pairs of repeated blocks if and only if \(V \equiv 0 \pmod{3}\), \[0\leq k \leq t_V = \frac{1}{6}V(V-3), \; \; k\neq t_V – 1, \text{and} (k,V)\neq(1,6)\].

For each integer \(v \geq 0\) and each \(\lambda \in \{4, 5, 7, 8\}\), the possible numbers of distinct blocks in a triple system of order \(v\) and index \(\lambda\) is determined. This essentially completes the determination of possible support sizes for triple systems with \(\lambda \leq 8\).

If \(n\) is an integer, \(n \geq 2\), and \(u\) and \(v\) are vertices of a graph \(G\), then \(u\) and \(v\) are said to be \(K_n\)-adjacent vertices of \(G\) if there is a subgraph of \(G\), isomorphic to \(K_n\), containing \(u\) and \(v\). A total \(K_n\)-dominating set of \(G\) is a set \(D\) of vertices such that every vertex of \(G\) is \(K_n\)-adjacent to a vertex of \(D\). The total \(K_n\)-domination number \(\gamma_{K_n}^t(G)\) of \(G\) is the minimum cardinality among the total \(K_n\)-dominating sets of vertices of \(G\). It is shown that, for \(n \in \{3, 4, 5\}\), if \(G\) is a graph with no \(K_n\)-isolated vertex, then \(\gamma_{K_n}^t(G) \leq (2p)/{n}\). Further, \(K_n\)-connectivity is defined and it is shown that, for \(n \in \{3, 4\}\), if \(G\) is a \(K_n\)-connected graph of order \(\geq n + 1\), then \(\gamma_{K_n}^t(G) \leq (2p)/(n + 1)\). We establish that the upper bounds obtained are best possible.

Let \(D\) and \(\overline{D}^d\) be two designs such that there is a joint embedding \(D’\) and \(\overline{D}’\) of \(D\) and \(\overline{D}\) in a finite projective plane \(\pi\) of order \(n\) such that the points of \(D’\) and the lines of \(\overline{D}’\) are mutually all of the exterior elements of each other. We show that there is a tactical decomposition \(T\) of \(\pi\), two of the tactical configurations of which are \(D’\) and \(\overline{D}’\), and determine combinatorial restrictions on \(n\) and the parameters of \(D\) and \(\overline{D}^d\). We also determine the entries of the incidence matrices of \(T\).

The Josephus problem is concerned with anticipating which will be the last elements left in the ordered set \(\{1, 2, \ldots, n\}\) as successive \(m\)th elements (counting cyclically) are eliminated. We study the set of permutations of \(\{1, 2, \ldots, n\}\) which arise from the different orders of elimination as \(m\) varies, and give a criterion based on the Chinese Remainder Theorem for deciding if a given permutation can be interpreted as arising as a given order of elimination for some step size \(m\) in a Josephus problem.