New identities involving the Catalan sequence ordinary generating function are developed, and a previously known one established from first principles using a hypergeometric approach.

We examine words \( w \) satisfying the following property: if \( x \) is a subword of \( w \) and \( |x| \) is at least \( k \) for some fixed \( k \), then the reversal of \( x \) is not a subword of \( w \).

For constructing routes in mobile ad-hoc networks (MANET) and sensor networks, it is highly desirable to perform primitive computations locally. If a network can be represented in the doubly connected edge list (DCEL) data structure, then many operations can be done locally. However, the DCEL data structure can be used to represent only planar graphs. In this paper, we propose an extended version of the DCEL data structure called ExtDCEL that can be used for representing non-planar graphs as well as their planar components. The proposed data structure can be used to represent geometric networks in mobile computing that include unit disk graphs, Gabriel graphs, and constrained Delaunay triangulations. We show how the proposed data structure can be used to implement a hybrid greedy face routing algorithm in optimum \( O(m) \) time, where \( m \) is the number of edges in the unit disk graph. We also report on the implementation of several routing algorithms for mobile computing by using the proposed data structure.

In this paper, we study the decomposition of the graph \( (\lambda D_v)^{+\alpha} \) into extended cyclic triples, for all \( \lambda \geq \alpha \). By an extended cyclic triple, we mean a loop, a loop with symmetric arcs attached (known as a lollipop), or a directed \( 3 \)-cycle (known as a cyclic triple).

In this paper, we consider the problem of the non-existence of some orthogonal arrays (O-arrays) of strength four with two levels, the number of constraints \( k \) satisfying \( 4 \leq k \leq 32 \), and index set \( \lambda \) where \( 1 \leq \lambda \leq 64 \).

We give a constructive proof that a planar graph on \( n \) vertices with degree of regularity \( k \) exists for all pairs \( (n,k) \) except for two pairs \( (7,4) \) and \( (14,5) \). We continue this theme by classifying all strongly regular planar graphs, and then consider a new class of graphs called \( 2 \)-\({strongly\; regular}\). We conclude with a conjectural classification of all planar \( 2 \)-strongly regular graphs.

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 \( n \geq 9 \).