A \((12,6,3)\) cover is a family of 6-element subsets, called blocks, chosen from a 12-element universe, such that each 3-element subset is contained in at least one block. This paper constructs a \((12,6,3)\) cover with 15 blocks, and it shows that any \((12,6,3)\) cover has at least 15 blocks; thus the covering number \(C(12,6,3) = 15\). It also shows that the 68 nonisomorphic \((12,6,3)\) covers with 15 blocks fall into just two classes using a very natural classification scheme.

An algorithm is given to generate all \(k\)-subsets of \(\{1, \ldots, n\}\) as increasing sequences, in an order so that going from one sequence to the next, exactly one entry is changed by at most \(2\).

Given a graph \(G\) with weighting \(w : E(G) \to \mathbb{Z}^+\), the strength of \(G(w)\) is the maximum weight on any edge. The weight of a vertex in \(G(w)\) is the sum of the weights of all its incident edges. The network \(G(w)\) is irregular if the vertex weights are distinct. The irregularity strength of \(G\) is the minimum strength of the graph under all irregular weightings. We determine the irregularity strength of the \(m \times n\) grid for all \(m, n \geq 18\).

The blocks of a balanced ternary design, \(\mathrm{BTD}(V, B; p_1, p_2, R; K, \Lambda)\), can be partitioned into two sets: the \(b_1\) blocks that each contain no repeated elements, and the \(b_2 = B – b_1\) blocks containing repeated elements. In this note, we address, and answer in some particular cases, the following question. For which partitions of the integer \(B\) as \(b_1 + b_2\) does there exist a \(\mathrm{BTD}(V, B; p_1, p_2, R; K, \Lambda)\)?

A general formula is obtained for the number of points lying on a plane algebraic curve over the finite local ring \(\mathrm{GF}(q)[t]/(t^n)\) (\(n > 1\)) whose equation has coefficients in \(\mathrm{GF}(q)\) and under the restriction that it has only simple and ordinary singular points.

Through combinatorial analysis we study the jump number, greediness and optimality of the products of chains, the product of an (upward rooted) tree and a chain. It is well known [1] that the dimension of products of \(n\) chains is \(n\). We construct a minimum realizer \(L_1, \ldots, L_n\) for the products of \(n\) chains such that \(s(\bigcap_{i=1}^{j}L_i) \leq s(\bigcap_{i=1}^{j+1}L_i)\) where \(j = 1, \ldots, n-1\).

In this paper, new optimal \((pm,m)\) and \((pm,m-1)\) ternary linear codes of dimension 6 are presented. These codes belong to the class of quasi-twisted codes, and have been constructed using a greedy local search algorithm. Other codes are also given which provide a lower bound on the maximum possible minimum distance. The minimum distances of known quasi-twisted codes of dimension 6 are given.

We propose the following conjecture: Let \(m \geq k \geq 2\) be integers such that \(k \mid m\), and let \(T_m\) be a tree on \(m\) edges. Let \(G\) be a graph with \(\delta(G) \geq m+k-1\). Then for every \(Z_k\)-colouring of the edges of \(G\) there is a zero-sum (mod \(k\)) copy of \(T_m\) in \(G\). We prove the conjecture for \(m \geq k = 2\), and explore several relations to the zero-sum Turán numbers.

For any double sequence \((q_{k,n})\) with \(q_{k,0} = 0\), the “summatorial sequence” \((P_{k,n}) = \sum(q_{k,n})\) is defined by \(p_{0,0} = 1\) and \(P_{k,} = \sum_{j=0}^k \sum_{m=1}^n q_{ j,m}P_{k-j,n-m}\) If \(q_{k,n} = 0\) for \(k < n-1\) then there exists a unique sequence \((c_j)\) satisfying the recurrence \(P_{k,n} = \sum_{j=0}^k c_j P_{k-j,k-j,n-m}\) for \(k < n\). We apply this combinatorial recursion to certain counting functions on finite posets. For example, given a set \(A\) of positive integers, let \(P_{k,n}\) denote the number of unlabeled posets with \(n\) points and exactly \(k\) antichains whose cardinality belongs to \(A\), and let \(q_{k,n}\) denote the corresponding number of ordinally indecomposable posets. Then \((P_{k,n})\) is the summatorial sequence of \((q_{k,n})\). If \(2 \in A\) then \((P_{k,n})\) enjoys the above recurrence for \(k < 1\). In particular, for fixed \(k\), there is a polynomial \(p_k\) of degree \(k\) such that \(P_{n,k} = p_k(n)\) for all \(n \geq k\), and \(p_{k,n}\) is asymptotically equal to \(\binom{n-1}{k}\). For some special classes \(A\) and small \(k\), we determine the numbers \(c_k\) and the polynomials \(p_k\) explicitly. Moreover, we show that, at least for small \(k\), the remainder sequences \(p_{k,n} – p_k(n)\) satisfy certain Fibonacci recursions, proving a conjecture of Culberson and Rawlins. Similar results are obtained for labeled posets and for naturally ordered sets.

The paper \([2]\) claimed that a disconnected graph with at least two nonisomorphic components is determined by some three of its vertex deleted subgraphs. While this statement is true, the proof in \([2]\) is incorrect. We give a correct proof of this fact.