Graph Lattices from Cycle-Guided Constructions of Edge-Count-Based Edge-Magic Total Labelings

Topological coding requires graph constructions that preserve structural and labeling constraints. Related methods attach trees to cycles or assemble bipartite components along trees. We develop an edge-count-based edge-magic total (EC-EMT) framework for connected bipartite graphs, allowing label gaps while maintaining total injectivity and a constant edge sum. Our main theorem combines components along a C4r skeleton, assuming a common edge count q and compatible set-ordered super EC-EMT labelings. The components may differ in order and structure and may themselves contain cycles. The output admits a set-ordered super EC-EMT labeling with 4r(q+1) edges and magic constant 10r(q+1)+4, extending cycle-based tree amalgamation to potentially multicyclic graphs. We establish transformations among set-ordered super EC-EMT, super odd EC-EMT, and super (k,d)-EC-EMT labelings, together with a double-copy edge-joining construction. Two leaf-addition results distinguish super-labeling preservation from coloring extensions with separate vertex and edge injectivity. Cycle composition and extremal leaf addition generate graph lattices recursively under explicit compatibility conditions. For supplied labeled inputs, construction takes linear time and space in the output size, assuming constant-time arithmetic. These results provide constructive tools for generating constrained graph families and their Topcode-matrix and number-based string representations.