A twofold 8-cycle system is an edge-disjoint decomposition of a twofold complete graph (which has two edges between every pair of vertices) into 8-cycles. The order of the complete graph is also called the order of the 8-cycle system. A twofold 2-perfect \(8\)-cycle system is a twofold \(8\)-cycle system such that the collection of distance \(2\) edges in each \(8\)-cycle also cover the complete graph, forming a (twofold) \(4\)-cycle system. Existence of \(2\)-perfect \(8\)-cycle systems for all admissible orders was shown in [1], although \(\lambda\)-fold existence for \(\lambda > 1\) has not been done.

In this paper, we impose an extra condition on the twofold \(2\)-perfect \(8\)-cycle system. We require that the two paths of length two between each pair of vertices, say \(x, a_{xy}, y\) and \(x, b_{xy}, y\), should be distinct, that is, with \(a_{xy} \neq b_{xy}\); thus they form a \(4\)-cycle \((x, a, y, b)\).

We completely solve the existence of such twofold \(2\)-perfect \(8\)-cycle systems with this “extra” property. All admissible orders congruent to \(0\) or \(1\) modulo \(8\) can be achieved, apart from order 8.

A graph \( G \) with \( k \) vertices is distance magic if the vertices can be labeled with numbers \( 1, 2, \ldots, k \) so that the sum of labels of the neighbors of each vertex is equal to the same constant \( \mu_0 \). We present a construction of distance magic graphs arising from arbitrary regular graphs based on an application of magic rectangles. We also solve a problem posed by Shafig, Ali, and Simanjuntak.

Given a graph \( G \), a function \( f : V(G) \to \{1, 2, \ldots, k\} \) is a \( k \)-ranking of \( G \) if \( f(u) = f(v) \) implies every \( u \)-\( v \) path contains a vertex \( w \) such that \( f(w) > f(u) \). A \( k \)-ranking is \emph{minimal} if the reduction of any label greater than \( 1 \) violates the described ranking property. The rank number of a graph, denoted \( \chi_r(G) \), is the minimum \( k \) such that \( G \) has a minimal \( k \)-ranking. The arank number of a graph, denoted \( \psi_r(G) \), is the maximum \( k \) such that \( G \) has a minimal \( k \)-ranking. It was asked by Laskar, Pillone, Eyabi, and Jacob if there is a family of graphs where minimal \( k \)-rankings exist for all \( \chi_r(G) \leq k \leq \psi_r(G) \). We give an affirmative answer showing that all intermediate minimal \( k \)-rankings exist for paths and cycles. We also give a characterization of all complete multipartite graphs which have this intermediate ranking property and which do not.

Let \( G \) be a \((p,q)\)-graph where each edge of \( G \) is labeled by a number \( 1, 2, \ldots, q \) without repetition. The vertex sum for a vertex \( v \) is the sum of the labels of edges that are incident to \( v \). If the vertex sums are equal to a constant (mod \( k \)) where \( k \geq 2 \), then \( G \) is said to be Mod(\( k \))-edge-magic. In this paper, we investigate graphs which are Mod(\( k \))-edge-magic. When \( k = p \), the corresponding Mod(\( p \))-edge-magic graph is the edge-magic graph introduced by Lee (third author), Seah, and Tan in \([10]\). In this work, we investigate trees, unicyclic graphs, and \((p, p+1)\)-graphs which are Mod(2)-edge-magic.

First posed in 1942 by Kelly and Ulam, the Graph Reconstruction Conjecture is one of the major open problems in graph theory. While the Graph Reconstruction Conjecture remains open, it has spawned a number of related questions. In the classical vertex graph reconstruction number problem, a vertex is deleted in every possible way from a graph \( G \), and then it can be asked how many (both minimum and maximum) of these subgraphs are required to reconstruct \( G \) up to isomorphism. This can then be extended to deleting \( k \) vertices in every possible way.

Previous computer searches have found the 1-vertex-deletion reconstruction numbers of all graphs of up to 11 vertices. In this paper, computed values of \( k \)-vertex-deletion reconstruction numbers for all graphs on up to 8 vertices and \( k \leq |V(G)| – 2 \) are reported, as well as for some \( k \) for graphs on 9 vertices. Our data suggested a number of new theorems and conjectures. In particular, we pose, as a generalization of the Graph Reconstruction Conjecture, that any graph on \( 3k \) or more vertices is \( k \)-vertex-deletion reconstructible.

Let \( G \) be a simple graph with a vertex set \( V(G) \) and an edge set \( E(G) \), and let \( \mathbb{Z}_2 = \{0,1\} \). A labeling \( f : V(G) \to \mathbb{Z}_2 \) induces an edge partial labeling \( f^* : E(G) \to \mathbb{Z}_2 \) defined by \( f^*(xy) = f(x) \) if and only if \( f(x) = f(y) \) for each edge \( xy \in E(G) \). For each \( i \in \mathbb{Z}_2 \), let \( v_f(i) = \lvert \{v \in V(G) : f(v) = i\} \rvert \) and \( e_f(i) = \lvert \{e \in E(G) : f^*(e) = i\} \rvert \). The balance index set of \( G \), denoted \( \text{BI}(G) \), is defined as \( \{\lvert e_f(0) – e_f(1) \rvert : \lvert v_f(0) – v_f(1) \rvert \leq 1\} \). In this paper, we investigate and present results concerning the balance index sets of trees of diameter four.