Finding the edge ranking number through vertex partitions

[[abstract]]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 chi(r)'(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 Sierpinski graph chi(r)'(S(n, k)) = n chi(r)'(K-k) for any n, k >= 2 where K-k denotes a complete graph of k vertices. (C) 2012 Elsevier B.V. All rights reserved.[[note]]SC