An edge coloring c′:E→{1,2,.,t} of a graph G=(V,E) is an edge t-ranking if for any two edges of the same color, every path between them contains an intermediate edge with a larger color. The edge ranking number χr′(G) is the smallest value of t such that G has an edge t-ranking. In this paper, we introduce a relation between edge ranking number and vertex partitions. By using the proposed recurrence formula, we show that the edge ranking number of the Sierpiński graph χr′(S(n,k))= nχr′(K_k) for any n,k≥2 where K_k denotes a complete graph of k vertices.
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