Finding the edge ranking number through vertex partitions

An edge coloring c′:E→{1,2,…,t}c′:E→{1,2,…,t} of a graph G=(V,E)G=(V,E) is an edge tt-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 tt such that GG has an edge tt-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′(Kk) for any n,k⩾2n,k⩾2 where KkKk denotes a complete graph of kk vertices.