Face Antimagic Labeling Techniques for House Graph Variants and Triangular Ladder Graphs

This study investigates the (𝒂, 𝒅)-face antimagic labeling on three classes of planar graphs, namely the house graph, the double house graph, and the triangular ladder graph. An (𝒂, 𝒅)-face antimagic labeling is defined as a bijective function from the set of vertices, edges, and faces to the set of positive integers, such that the sum of the labels assigned to the vertices and edges incident with each face forms an arithmetic sequence. In this work, specific labeling constructions are developed for the vertices, edges, and faces of each graph to ensure that the face antimagic property is satisfied. Illustrative examples are presented to demonstrate the labeling schemes that produce face weights forming an arithmetic sequence with a fixed common difference of 𝒅 = 𝟏. The findings confirm that all three classes of graphs admit valid face antimagic labelings. This research contributes to the advancement of graph labeling theory, particularly in the study of structured planar graphs