We consider square arrays of numbers \(\{a(n, k)\}\), generalizing the binomial coefficients:
\(a(n, 0) = c_n\), where the \(c_n\) are non-negative real numbers; \(a(0, k) = c_0\), and if \(n, k > 0\), then \(a(n, k) = a(n, k – 1) + a(n – 1, k)\).
We give generating functions and arithmetical relations for these numbers. We show that every row of such an array is eventually log concave, and give a few sufficient conditions for columns to be eventually log concave. We also give a necessary condition for a column to be eventually log concave, and provide examples to show that there exist such arrays in which no column is eventually log concave.

In this paper, we obtain some necessary conditions for the existence of balanced arrays (\(B\)-arrays) of strength \(4\) and with two levels, and we state the usefulness of these conditions in obtaining an upper bound on the number of constraints for these B-arrays.

It is shown that the circuit polynomial of a graph, when weighted by the number of nodes in the circuits, does not characterize the graph, i.e., non-isomorphic graphs can have the same circuit polynomial. Some general theorems are given for constructing graphs with the same circuit polynomial (cocircuit graphs). Analogous results can be deduced for characteristic polynomials.

Given \(n\) real numbers whose sum is zero, find one of the numbers that is non-negative. In the model under consideration, an algorithm is allowed to compute \(p\) linear forms in each time step until it knows an answer. We prove that exactly \(\lceil{\log n}/{\log(p+1)} \rceil\) time steps are required. Some connections with parallel group-testing problems are pointed out.

With the help of a computer, the third Ramsey number is determined for each of the \(25\) graphs with five edges, five or more vertices, and no trivial components.

In this paper, we prove that the size Ramsey number

\[\hat{r}(a_1K_{1,n_1}, a_2K_{1,n_2}, \ldots ,a_\ell K_{1,n_\ell}) = \left[\sum\limits_{i=1}^\ell {(a_i – 1)+1} \right] \left[\sum\limits_{i=1}^\ell {(n_i – 1)+1} \right].\]

We consider the following three problems: Given a graph \(G\),

1. What is the smallest number of cliques into which the edges of \(G\) can be partitioned?
2. How many cliques are needed to cover the edges of \(G\)?
3. Can the edges of \(G\) be partitioned into maximal cliques of \(G\)?

All three problems are known to be NP-complete for general \(G\). We show here that: (1) is NP-complete for \(\Delta(G) \geq 5\), but can be solved in polynomial time if \(\Delta(G) \leq 4\) (the latter has already been proved by Pullman \([P]\)); (2) is NP-complete for \(\Delta(G) \geq 6\), and polynomial for \(\Delta(G) \leq 5\); (3) is NP-complete for \(\Delta(G) \geq 8\) and polynomial time for \(\Delta(G) \leq 7\).

The total chromatic number \(\chi_2(G)\) of a graph \(G\) is the smallest number of colors which can be assigned to the vertices and edges of \(G\) so that adjacent or incident elements are assigned different colors. For a positive integer \(m\) and the star graph \(K_{1,n}\), the mixed Ramsey number \(\chi_2(m, K_{1,n})\) is the least positive integer \(p\) such that if \(G\) is any graph of order \(p\), either \(\chi_2(G) \geq m\) or the complement \(\overline{G}\) contains \(K_{1,n}\) as a subgraph.
In this paper, we introduce the concept of total chromatic matrix and use it to show the following lower bound: \(\chi_2(m, K_{1,n}) \geq m + n – 2\) for \(m \geq 3\) and \(n \geq 1\). Combining this lower bound with the known upper bound (Fink), we obtain that \(\chi_2(m, K_{1,n}) = m + n – 2\) for \(m\) odd and \(n\) even, and \(m + n – 2 \leq \chi_2(m, K_{1,n}) \leq m + n – 1\) otherwise.

An addition-multiplication magic square of order \(n\) is an \(n \times n\) matrix whose entries are \(n^2\) distinct positive integers such that not only the sum but also the product of the entries in each row, column, main diagonal, and back diagonal is a constant. It is shown in this paper that such a square exists for any order \(mn\), where \(m\) and \(n\) are positive integers and \(m, n \notin \{1, 2, 3, 6\}\).

A hypergraph is irregular if no two of its vertices have the same degree. It is shown that for all \(r \geq 3\) and \(n \geq r + 3\), there exist irregular \(r\)-uniform hypergraphs of order \(n\). For \(r \geq 6\) it is proved that almost all \(r\)-uniform hypergraphs are irregular. A linear upper bound is given for the irregularity strength of hypergraphs of order \(n\) and fixed rank. Furthermore, the irregularity strength of complete and complete equipartite hypergraphs is determined.