MAKASSAR, Indonesia March 3 2026/graphtheory/ Imagine several circular routes meeting at one common hub. From a distance, the structure looks simple, but identifying every point inside it can be surprisingly difficult. This is the challenge explored in the study “The Partition Dimension of the Vertex Amalgamation of Some Cycles.”
The research belongs to graph theory, a branch of mathematics used to study networks. In a graph, points are called vertices and the links between them are called edges. A cycle graph is a closed loop, similar to a circular road. Vertex amalgamation means joining several cycle graphs at one shared vertex, creating a flower-like network with one central meeting point.
The main question is: how can every vertex in this network be uniquely identified using only its distances from several groups of vertices?
To answer this, the researchers use the concept of partition dimension. First, all vertices are divided into several groups, called partition classes. For each vertex, its shortest distance to every group is calculated. These distances form a numerical address. A partition works when no two vertices have the same address. The partition dimension is the smallest number of groups needed to distinguish every vertex.
This idea is similar to locating houses in a complex neighborhood. A house may be described by its distance from a school, market, hospital, and station. If every house receives a different combination of distances, each one can be identified without using a conventional street address.
The study focuses on networks formed by combining many copies of the same cycle at a single center. The researchers show that the required partition dimension increases in a predictable way as more cycles are attached. For a given integer k, when the number of cycles m lies between
the partition dimension of the amalgamated cycle graph is exactly k, under the conditions examined in the article. This means the researchers established a clear mathematical rule rather than relying on repeated trial and error.
The diagram in the article shows seven cycle graphs joined at one central vertex. Although many vertices appear symmetrical, carefully designed partition classes produce different distance patterns for them. The study also explains why vertices occupying similar positions in different cycles can be difficult to distinguish and therefore must be separated through an appropriate partition strategy.
This research directly supports Sustainable Development Goal 4: Quality Education. It expands knowledge in discrete mathematics, provides a structured theorem for teaching graph theory, and offers a useful example of how abstract reasoning can solve identification problems in complex networks. The same ideas may support future studies in navigation, communication networks, robotics, and sensor placement.
Ultimately, the study shows that distance can function like an address. Even in a highly symmetrical network of joined circles, mathematics can determine the smallest amount of information needed to tell every point apart.
Reference:
DOI: https://doi.org/10.1016/j.heliyon.2022.e09596
Contact:
Prof. Nurdin
+62 813-9519-0801
nurdin1701@unhas.ac.id
