Adjacent Vertex Reducible Edge Labeling Algorithm for Fan Graphs

For a simple undirected connected graph, if there exists a one-to-one mapping of the set of edges, which is labeled and the same for all adjacent vertices of the same degree in the graph, then it is called Adjacent Vertex Reducible Edge Labeling (AVREL) of the graph. Aiming at the problem of AVREL, a recursive search algorithm is designed, which is used to label the fan graphs, to obtain the labeling laws of the fan graphs and the union graphs, and to summarize the relevant theorems and give the proofs by combining the existing theorems.