Let \( G = (V(G), E(G)) \) be a simple, finite, and undirected graph with \( n \) vertices. Given a bijection \( f : V(G) \to \{1, \dots, n\} \), one can associate two integers \( S = f(u) + f(v) \) and \( D = |f(u) – f(v)| \) with every edge \( uv \in E(G) \). The labeling \( f \) induces an edge labeling \( f’ : E(G) \to \{0, 1\} \) such that for any edge \( uv \) in \( E(G) \), \( f'(uv) = 1 \) if \(\gcd(S, D) = 1\), and \( f'(uv) = 0 \) otherwise. Such a labeling is called an SD-prime labeling if \( f'(uv) = 1 \) for all \( uv \in E(G) \). We say that \( G \) is SD-prime if it admits an SD-prime labeling. A graph \( G \) is said to be a \({strongly \;SD-prime \;graph}\) if for every vertex \( v \) of \( G \) there exists an SD-prime labeling \( f \) satisfying \( f(v) = 1 \). In this paper, we first give some sufficient conditions for a theta graph to be strongly SD-prime. We then provide constructions of new SD-prime graphs from known SD-prime graphs and investigate the SD-primality of some general graphs.

Recently, the authors proposed a fundamental theorem for the decomposing of a complete bipartite graph. They applied the theorem to obtain complete results on the decomposition of a complete bipartite graph into connected subgraphs on four vertices and up to four edges. In this paper, we decompose a complete multi-bipartite graph into its subgraphs of four vertices and five edges. We show that necessary conditions are sufficient for the decompositions, with some exceptions where decompositions do not exist

The authors previously defined the Stanton-type graph \(S(n,m)\) and demonstrated how to decompose \(\lambda K_n\) (for the appropriate minimal values of \(\lambda\)) into Stanton-type graphs \(S(4, 3)\) of the LOE-, OLE-, LEO-, and ELO-types. Sarvate and Zhang showed that for all possible values of \(\lambda\), the necessary conditions are sufficient for LOE- and OLE-decompositions. In this paper, we show that for all possible values of \(\lambda\), the necessary conditions are sufficient for LEO- and ELO-decompositions.

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). A \((p, q)\)-graph \(G = (V, E)\) is said to be \(AL(k)\)-traversal if there exists a sequence of vertices \(\{v_1, v_2, \ldots, v_p\}\) such that for each \(i = 1, 2, \ldots, p-1\), the distance between \(v_i\) and \(v_{i+1}\) is equal to \(k\). We call a graph \(G\) a \(k\)-steps Hamiltonian graph if it has an \(AL(k)\)-traversal in \(G\) and the distance between \(v_p\) and \(v_1\) is \(k\). In this paper, we completely classify whether a subdivision graph of a cycle with a chord is \(2\)-steps Hamiltonian.

A triple system is decomposable if the blocks can be partitioned into two sets, each of which is itself a triple system. It is cyclically decomposable if the resulting triple systems are themselves cyclic. In this paper, we prove that a cyclic two-fold triple system is cyclically indecomposable if and only if it is indecomposable. Moreover, we construct cyclic three-fold triple systems of order $v$ which are cyclically indecomposable but decomposable for all \(v \equiv 3 \mod 6\). The only known example of a cyclic three-fold triple system of order \(v \equiv 1 \mod 6\) that is cyclically indecomposable but decomposable was a triple system on 19 points. We present a construction which yields infinitely many such triple systems of order \(v \equiv 1 \mod 6\). We also give several examples of cyclically indecomposable but decomposable cyclic four-fold triple systems and few constructions that yield infinitely many such triple systems.

The spectrum problem for decomposition of trees with up to eight edges was introduced and solved in 1978 by Huang and Rosa. Additionally, the packing problem was settled for all trees with up to six edges by Roditty. For the first time, we consider obtaining all possible leaves in a maximum tree-packing of \(K_n\), which we refer to as the spectrum problem for packings of complete graphs. In particular, we completely solve this problem for trees with at most five edges. The packing designs are used in developing optimal error-correcting codes, which have applications in biology, such as in DNA sequencing.

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). A labeling \(f\) of a graph \(G\) is said to be edge-friendly if \(|e_f(0) – e_f(1)| \leq 1\), where \(e_f(i) = \text{card}\{e \in E(G) : f(e) = i\}\). An edge-friendly labeling \(f : E(G) \to \mathbb{Z}_2\) induces a partial vertex labeling \(f^+ : V(G) \to A\) defined by \(f^+(x) = 0\) if the edges incident to \(x\) are labeled \(0\) more than \(1\). Similarly, \(f^+(x) = 1\) if the edges incident to \(x\) are labeled \(1\) more than \(0\). \(f^+(x)\) is not defined if the edges incident to \(x\) are labeled \(1\) and \(0\) equally. The edge-balance index set of the graph \(G\), \(EBI(G)\), is defined as \(\{|v_f(0) – v_f(1)| : \text{the edge labeling } f \text{ is edge-friendly}\}\), where \(v_f(i) = \text{card}\{v \in V(G) : f^+(v) = i\}\).

An \(n\)-wheel is a graph consisting of \(n\) cycles, with each vertex of the cycles connected to one central hub vertex. The edge-balance index sets of \(n\)-wheels are presented.

A set of vertices \(W\) \({locally\; resolves}\) a graph \(G\) if every pair of adjacent vertices is uniquely determined by its coordinate of distances to the vertices in \(W\). The minimum cardinality of a local resolving set of \(G\) is called the \({local\; metric\; dimension}\) of \(G\). A graph \(G\) is called a \(k\)-regular graph if every vertex of \(G\) is adjacent to \(k\) other vertices of \(G\). In this paper, we determine the local metric dimension of an \((n-3)\)-regular graph \(G\) of order \(n\), where \(n \geq 5\).

Let \(\mathcal{G}_n\) be the set of all simple loopless undirected graphs on \(n\) vertices. Let \(T\) be a linear mapping, \(T: \mathcal{G}_n \to \mathcal{G}_n\), such that the dot product dimension of \(T(G)\) is the same as the dot product dimension of \(G\) for any \(G \in \mathcal{G}_n\). We show that \(T\) is necessarily a vertex permutation. Similar results are obtained for mappings that preserve sets of graphs with specified dot product dimensions.

A permutation \(\pi\) on a set of positive integers \(\{a_1, a_2, \ldots, a_n\}\) is said to be graphical if there exists a graph containing exactly \(a_i\) vertices of degree \(\pi(a_i)\) for each \(i\) (\(1 \leq i \leq n\)). It has been shown that for positive integers with \(a_1 < a_2 < \ldots < a_n\), if \(\pi(a_n) = a_n\), then the permutation \(\pi\) is graphical if and only if the sum \(\sum_{i=1}^n a_i \pi(a_i)\) is even and \(a_n \leq \sum_{i=1}^{n-1} a_i\pi(a_i)\).

We use a criterion of Tripathi and Vijay to provide a new proof of this result and to establish a similar result for permutations \(\pi\) such that \(\pi(a_{n-1}) = a_n\). We prove that such a permutation is graphical if and only if the sum \(\sum_{i=1}^n a_i \pi(a_i)\) is even and \(a_na_{n-1} \leq a_{n-1}(a_{n-1} – 1) + \sum_{i\neq n-1} a_i\pi(a_i)\). We also consider permutations such that \(\pi(a_n) = a_{n-1}\) and, more generally, those such that \(\pi(a_n) = a_{n-j}\) for some \(j\) (\(1 < j < n\)).