In this paper, the notions of \(c\)-Motzkin and \(d\)-Motzkin words are introduced, studied, and the cardinal numbers of their sets are evaluated. Finally, bijections between the sets of the introduced Motzkin words and certain sets of noncrossing partitions are exhibited.

Vizing conjectured that \(\gamma(G)\gamma(H) \leq \gamma(G \Box H)\) for all graphs \(G\) and \(H\), where \(\gamma(G)\) denotes the domination number of \(G\) and \(G \Box H\) is the Cartesian product of \(G\) and \(H\). We prove that if \(G\) and \(H\) are \(\delta\)-regular, then, with only a few possible exceptions, Vizing’s conjecture holds. We also prove that if \(\delta(G), \Delta(G), \delta(H)\), and \(\Delta(H)\) are in a certain range, then Vizing’s conjecture holds. In particular, we show that for graphs of order at most \(n\) with minimum degrees at least \(\sqrt{n} \ln n\), the conjecture holds.

The classification of Hadamard matrices of orders \(n \geq 32\) remains an open and difficult problem. The definition of equivalent Hadamard matrices gets increasingly complex as \(n\) grows larger. One efficient criterion (\(K\)-boxes) has been used for the construction of inequivalent Hadamard matrices in order \(28\).

In this paper, we use inequivalent projections of Hadamard matrices and their symmetric Hamming distances to check for inequivalent Hadamard matrices. Using this criterion, we have developed two algorithms. The first one achieves finding all inequivalent projections in \(k\) columns as well as classifying Hadamard matrices, and the second, which is faster than the first, uses the symmetric Hamming distance distribution of projections to classify Hadamard matrices. As an example, we apply the second algorithm to the known inequivalent Hadamard matrices of orders \(n = 4, 8, 12, 16, 20, 24\), and \(28\).

A composition of a positive integer \(n\) consists of an ordered sequence of positive integers whose sum is \(n\). A palindromic composition is one for which the sequence is the same from left to right as from right to left. This paper shows various ways of generating all palindromic compositions, counts the number of times each integer appears as a summand among all the palindromic compositions of \(n\), and describes several patterns among the numbers generated in the process of enumeration.

The study of the maximum size \(ex(n; K_{t,t})\) of a graph of order \(n\) not containing the complete bipartite graph \(K_{t,t}\) as a subgraph is the aim of this paper. We show an upper bound for this extremal function that is optimum for infinitely many values of \(n\) and \(t\). Moreover, we characterize the corresponding family of extremal graphs.

In this paper we extend the work of Bogart and Trenk [3] and Fishburn and Trotter [6] in studying different classes of bitolerance orders. We provide a more comprehensive list of classes of bitolerance orders and prove equality between some of these classes in general and other classes in the bipartite domain. We also provide separating examples between unequal classes of bitolerance orders.

We consider non-crossing trees and show that the height of node \(\rho n\) with \(0 < p < 1\) in a non-crossing tree of size \(n\) is asymptotically Maxwell-distributed. We also give an asymptotic formula for the expected height of node \(\rho n\).

Let \(G = (V(G), E(G))\) be a finite simple graph with \(p\) vertices and \(n\) edges. A labeling of \(G\) is an injection \(f: V(G) \to {Z}_n\). A labeling of \(G\) is called \(2\)-sequential if \(f(V(G)) = \{r, r+1, \ldots, r+p-1\}\) (\(0 \leq r <r+ p-1 \leq n-1\)) and the induced edge labeling \(f^*: E(G) \to \{0, 1, \ldots, n-1\}\) given by \[f^*(u,v) = f(u) + f(v), \quad \text{for every edge } (u,v) \] forms a sequence of distinct consecutive integers \(\{k, k+1, \ldots, n+k-1\}\) for some \(k\) (\(1 \leq k \leq n-2\)). By utilizing the graphs having \(2\)-sequential labeling, several new families of sequential graphs are presented.

A cycle \(C\) in a graph \(G\) is said to be a dominating cycle if every vertex of \(G\) has a neighbor on \(C\). Strengthening a result of Bondy and Fan [3] for tough graphs, we prove that a \(k\)-connected graph \(G\) (\(k \geq 2\)) of order \(p\) with \(t(G) > \frac{k}{k+1}\) has a dominating cycle if \(\sum_{x \in S} \geq p – 2k – 2\) for each \(S \subset V(G)\) of order \(k+1\) in which every pair of vertices in \(S\) have distance at least four in \(G\).

Let \(G = (V,E)\) be an n-vertex graph and \(f : V \rightarrow \{1,2,\ldots,n\}\) be a bijection. The additive bandwidth of \(G\), denoted \(B^+(G)\), is given by \(B^+(G) = \min_{f} \max_{u,v\in E} |f(u) + f(v) – (n+1)|\), where the minimum ranges over all possible bijections \(f\). The additive bandwidth cannot decrease when an edge is added, but it can increase to a value which is as much as three times the original additive bandwidth. The actual increase depends on \(B^+(G)\) and n and is completely determined.