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 \(G\).