On Even Vertex Magic Total Labelings of Plus Wheels and Some Wheel-Related Graphs

Let G be a graph with n vertices and m edges. A vertex magic total labeling of G is a bijection f:V(G)∪E(G)→{1,2,…,n+m} such that, for each vertex u∈V(G), the sum of the label of u and the labels of all edges incident to u is equal to a fixed constant, referred to as the magic constant. A vertex magic total labeling is said to be even if the labels assigned to the vertices are exactly even numbers {2,4,6,…,2n}. These labelings, along with related variations, have theoretical significance and practical applications, such as resource allocation, fault tolerance, and network design. Structured labelings aid channel assignment, address computation, and reduce collisions in networks. In this paper, we investigate wheel-related graphs that either admit or do not admit an even vertex magic total labeling. Furthermore, we introduce a new class of wheel-related graph, referred to as the plus wheel Wn+r, that can have such labelings, and we also establish a necessary and sufficient condition for such graphs to possess this property.