On the basis of power-cycle nested network graph, the edge-balanced index sets of circle union graph F(3,n) graph were investigated. A new method of changing index is provided, simplifying the proving process. It reduced the difficulty of circle union graph F(3,n) graph labeling because of the novel design of the basic graph and single-point sector subgraph. The results show that the edge-balanced index sets of circle union graph F(3,n) graph. This paper has proved the existence of the edge-balanced index sets of one class of circle union graph, the computational formulas and the construction of the corresponding graphs are also provided.