The Relationship of D(β)-Vertex Sum Distinguishing Edge Chromatic Numbers Between Graphs and Sub-graphs

Using the method of constructive coloring and combinatorial analysis, the relationship between the D(β)-vertex sum distinguishing edge chromatic numbers of a simple connected graph G and its sub-graphs is investigated where the distance between any two vertices in G does not exceed 2.The results show that not all the D(β)-vertex sum distinguishing edge chromatic number of graph are greater than those of its sub-graphs, and there exist cases where the D(β)-vertex sum distinguishing edge chromatic number of sub-graphs is greater than that of the graph G.