For a graph \(G\), let \(\mathcal{D}(G)\) be the set of all strong orientations of \(G\). Define the orientation number of \(G\), \(\overrightarrow{d}(G) = \min\{d(D) \mid D \in \mathcal{D}(G)\}\), where \(d(D)\) denotes the diameter of the digraph \(D\). In this paper, it has been shown that \(\overrightarrow{d}(G \times H) = d(G)\), where \(\times\) denotes the tensor product of graphs, \(H\) is a special type of circulant graph, and the diameter, \(d(G)\), of \(G\) is at least \(4\). Some interesting results have been obtained using this result. Further, it is shown that \(d(P_r \times K_s) = d(P_r)\) for suitable \(r\) and \(s\). Moreover, it is proved that \(\overrightarrow{d}(C_r \times K_s) = d(C_r)\) for appropriate \(r\) and \(s\).

We consider some partitions where even parts appear twice and some where evens do not repeat. Further, we offer a new partition theoretic interpretation of two mock theta functions of order \(8\).

A graph is said to be cordial if it has a \(0-1\) labeling that satisfies certain properties. The purpose of this paper is to generalize some known theorems and results of cordial graphs. Specifically, we show that certain combinations of paths, cycles, and stars are cordial.

An edge-magic total labeling on a graph with \(p\) vertices and \(q\) edges is defined as a one-to-one map taking the vertices and edges onto the integers \(1, 2, \ldots, p+q\) with the property that the sum of the labels on an edge and of its endpoints is constant, independent of the choice of edge. The magic strength of a graph \(G\), denoted by \(emt(G)\), is defined as the minimum of all constants over all edge-magic total labelings of \(G\). The maximum magic strength of a graph \(G\), denoted by \(eMt(G)\), is defined as the maximum constant over all edge-magic total labelings of \(G\). A graph \(G\) is called weak magic if \(eMt(G) – emt(G) > p\). In this paper, we study some classes of weak magic graphs.

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.