The Non-Edge-Magic Total Labeling Algorithm for Several Join Graphs

For the undirected connected graph G(p, q), if there exists a mapping f: V (G)∪E(G) →{1,2,…, p+q} such that for any edge V (G)∪E (G), there exists f (u)+ f (v)+ f (uv) = k, with k being a constant, then the graph G(p, q) is an edge-magic graph; otherwise, the graph is a non-edge-magic total labeling graph. This paper designs a new heuristic search algorithm aimed at identifying and determining whether a given graph is a non-edge-magic graph (NEMTL graph). By analyzing the algorithm’s results on the labeling patterns of non-edge-magic graphs within a finite number of vertices, it summarizes several theorems on join graphs and proves, through the introduction of Sidon sequences, that when m and n meet certain conditions, the join graphs K
n
↑ C
m
and K
n
↑ F
m
possess non-edge-magic properties, where the graph operation is denoted by ↑.