In the first part of this paper, we present a generalization of complete graph factorizations obtained by labeling the graph vertices by natural numbers. In this generalization, the vertices are labeled by elements of an arbitrary group \(G\), in order to achieve a \(G\)-transitive factorization of the graph.

Vertex colorings of Steiner systems \(S(t,t+1,v)\) are considered in which each block contains at least two vertices of the same color. Necessary conditions for the existence of such colorings with given parameters are determined, and an upper bound of the order \(O(\ln v)\) is found for the maximum number of colors. This bound remains valid for nearly complete partial Steiner systems, too. In striking contrast, systems \(S(t,k,v)\) with \(k \geq t+2\) always admit colorings with at least \(c\cdot v^\alpha\) colors, for some positive constants \(c\) and \(\alpha\), as \(v\to\infty\).

Cwatsets were originally defined as subsets of \(\mathbb{Z}_2^d\) that are “closed with a twist.” Attempts have been made to generalize them, but the generalizations have failed to produce notions of subcwatset and quotient cwatset that behave naturally.

We present a new, abstract definition that appears to avoid these problems. The relationship between this new definition and its predecessor is similar to that between the abstract definition of “group” and its original meaning as a set of permutations. To justify the broader definition, we use small cancellation theory to prove a result analogous to the statement that every group is isomorphic to some permutation group. After developing the notion of a quotient cwatset, we prove an analogue of the First Homomorphism Theorem.

In this paper, we consider a class of recursively defined, full binary trees called Lucas trees and investigate their basic properties. In particular, the distribution of leaves in the trees will be carefully studied. We then go on to show that these trees are \(2\)-splittable, i.e., they can be partitioned into two isomorphic subgraphs. Finally, we investigate the total path length and external path length in these trees, the Fibonacci trees, and other full \(m\)-ary trees.

A tree \(T\) with \(n\) vertices and a perfect matching \(M\) is strongly graceful if \(T\) admits a graceful labeling \(f\) such that \(f(u)+f(v) = n-1\) for every edge \(uv \in M\). Broersma and Hoede \([5]\) conjectured that every tree containing a perfect matching is strongly graceful in \(1999\). We prove that a tree \(T\) with diameter \(D(T) \leq 5\) supports the strongly graceful conjecture on trees. We show several classes of basic seeds and some constructive methods for constructing large scales of strongly graceful trees.

In a previous paper, the first author introduced two classes of generalized Stirling numbers, \(s_m(n,k,p), S_m(n,k,p)\) with \(m = 1\) or \(2\), called \(p\)-Stirling numbers. In this paper, we discuss their determinant properties.

The Padmakar-Ivan (PI) index is a Wiener-Szeged-like topological index which reflects certain structural features of organic molecules. In this paper, we study the problem of PI index with respect to some simple pericondensed hexagonal systems and we solve it completely.

As a part of the author’s work of enumerating the edge-forwarding indices of Frobenius graphs, I give a class of valency four Frobenius graphs derived from the Frobenius groups \(\mathbb{Z}_{4n^2+1} \rtimes \mathbb{Z}_4\). Following the method of Fang, Li and Praeger, some properties including the diameter and the type of this class of graphs are given (Theorem \(3.2\)).

We address the problem of determining all sets which form minimal covers of maximal cliques for interval graphs. We produce an algorithm enumerating all minimal covers using the C-minimal elements of the interval order, as well as an independence Metropolis sampler. We characterize maximal removable sets, which are the complements of minimal covers, and produce a distinct algorithm to enumerate them. We use this last characterization to provide bounds on the maximum number of minimal covers for an interval order with a given number of maximal cliques, and present some simulation results on the number of minimal covers in different settings.

A directed triple system of order \(v\), denoted by DTS\((v)\), is a pair \((X,\mathcal{B})\) where \(X\) is a \(v\)-set and \(\mathcal{B}\) is a collection of transitive triples on \(X\) such that every ordered pair of \(X\) belongs to exactly one triple of \(\mathcal{B}\). A DTS\((v)\) is called pure and denoted by PDTS\((v)\) if \((x,y,z) \in \mathcal{B}\) implies \((z,y,x) \notin \mathcal{B}\). A large set of disjoint PDTS\((v)\) is denoted by LPDTS\((v)\). In this paper, we establish the existence of LPDTS\((v)\) for \(v \equiv 0,4 \pmod{6}\), \(v\geq 4\).