This paper answers the question as to whether every natural number \( n \) is realizable as the number of ones in the top portion of rows of a general binary Pascal triangle. Moreover, the minimum number \( \kappa(n) \) of rows is determined so that \( n \) is realizable.

A \( (p,q) \)-graph \( G \) is said to be \textbf{edge-graceful} if the edges can be labeled by \( 1,2,\ldots, q \) so that the vertex sums are distinct, mod \( p \). It is shown that if a tree \( T \) is edge-graceful, then its order must be odd. Lee conjectured that all trees of odd orders are edge-graceful. The conjecture is still unsettled. In this paper, we give the state of the progress toward this tantalizing conjecture.

We use a new technique for decomposition of complete graphs with even number of vertices based on \( 2n \)-cyclic blended labeling to show that for every \( k > 1 \) odd, and every \( d \), \( 3 \leq d \leq 2^qk – 1 \), there exists a spanning tree of diameter \( d \) that factorizes \( K_{2^qk} \).

A constant composition code of length \( n \) over a \( k \)-ary alphabet has the property that the numbers of occurrences of the \( k \) symbols within a codeword is the same for each codeword. These specialize to constant weight codes in the binary case, and permutation codes in the case that each symbol occurs exactly once. Constant composition codes arise in powerline communication and balanced scheduling, and are used in the construction of permutation codes. Using exhaustive and probabilistic clique search, and by applying theorems and constructions in past literature, we generate tables which summarize the best known lower bounds on constant composition codes for (i) \( 3 \leq k \leq 8 \), (ii) \( k = 3 \), \( 9 \leq n \leq 12 \), and (iii) various other interesting parameters with \( n \geq 9 \).

In this paper, we develop a computational method for constructing transverse \( t \)-designs. An algorithm is presented that computes the \( G \)-orbits of \( k \)-element subsets transverse to a partition \( \mathcal{H} \), given that an automorphism group \( G \) is provided. We then use this method to investigate transverse Steiner quadruple systems. We also develop recursive constructions for transverse Steiner quadruple systems, and we provide a table of existence results for these designs when the number of points \( v \leq 24 \). Finally, some results on transverse \( t \)-designs with \( t > 3 \) are also presented.

A vertex-magic total labeling of a graph \( G(V, E) \) is defined as a one-to-one mapping from \( V \cup E \) to the set of integers \( \{1,2,\ldots,|V| + |E|\} \) with the property that the sum of the label of a vertex and the labels of all edges incident to this vertex is the same constant for all vertices of the graph. A supermagic labeling of a graph \( G(V, E) \) is defined as a one-to-one mapping from \( E \) to the set of integers \( \{1, 2,\ldots,|E|\} \) with the property that the sum of the labels of all edges incident to a vertex is the same constant for all vertices of the graph.

In this paper, we present a technique for constructing vertex-magic total labelings of products of certain vertex-magic total \( r \)-regular graphs \( G \) and certain \( 2_s \)-regular supermagic graphs \( H \). \( H \) has to be decomposable into two \( s \)-regular factors and if \( r \) is even, \( |H| \) has to be odd.

At each vertex in a Cayley map, the darts emanating from that vertex are labeled by a generating set of a group. This generating set is closed under inverses. Two classes of Cayley maps are balanced and antibalanced maps. For these cases, the distributions of the inverses about the vertex are well understood. For a balanced Cayley map, either all the generators are involutions or each generator is directly opposite across the vertex from its inverse. For an antibalanced Cayley map, there is a line of reflection in the tangent plane of the vertex so that the inverse generator for each dart label is symmetric across that line. An \( e \)-balanced Cayley map is a recent generalization that has received much study, see for example [2, 6, 7, 13]. In this note, we examine the symmetries of the inverse distributions of \( e \)-balanced maps in a manner analogous to those of balanced and antibalanced maps.