Edge-Magic Total Labelling of Cyclic and Bicyclic Bridge Graphs

Abstract

Edge-magic total labelling is an interesting area in graph theory with significant implications. In this study, we explore the edge-magic total labelling of cyclic graphs with n vertices and bicyclic bridge graphs with 2n vertices, demonstrating that these graphs can be labelled with a magic sum k=2n. An edge- magic total labelling on a graph G is a one-to-one map 𝝀 from 𝑽(𝑮) ∪𝑬(𝑮) onto the integers 1,2,…,𝒗 + 𝒆, where 𝒗 = |𝑽(𝑮)| and 𝒆 = |𝑬(𝑮)|. This mapping has the property that for any edge 𝒙𝒚, 𝝀(𝒙) + 𝝀(𝒙𝒚) + 𝝀(𝒚) = 𝒌, a constant called the magic sum of 𝑮. Graphs that satisfy this condition are termed edge-magic. For cyclic graphs with 𝒏 vertices, we start by labelling the vertices from 𝟏 to 𝒏 in a clockwise direction. Edges are then labelled by starting from the (𝒏 − 𝟏)th edge, labelling from 𝟏 to 𝟐𝒏−𝟑 in steps of 𝟐 in an anti-clockwise direction, and the � �th edge is labelled 𝒏 − 𝟏. Considering any edge 𝒙𝒚 with adjacent vertices labelled 𝒎 + 𝟏 and 𝒎, the edge receives the label 𝟐𝒏 − 𝟐𝒎−𝟏. The magic sum 𝒌 is calculated as 𝒎 + (𝒎+𝟏)+𝟐(𝒏−𝒎)−𝟏=𝟐𝒏, proving that cyclic graphs with n vertices are edge-magic with the magic sum 𝟐𝒏. For bicyclic bridge graphs, two cyclic graphs each with 𝒏 vertices are connected by a bridge. Each cycle is labelled similarly to the cyclic graph. The bridge connects the vertex labelled 𝟏 of each cycle and is labelled 𝟐𝒏 − 𝟐. For the bridge edge, the magic sum remains 𝟐𝒏. Thus, the bicyclic bridge graphs are also edge-magic with the magic sum 𝟐𝒏. This study confirms that both cyclic graphs with 𝒏 vertices and bicyclic bridge graphs with 𝟐𝒏 vertices can achieve edge-magic total labelling with a consistent magic sum of 𝟐𝒏, contributing to the broader understanding of labelling in graph theory.