An r-maximum k-edge-coloring of G is a k-edge-coloring of G having a property that for every vertex v of degree dG(v) = d, d ≥ r, the maximum color, that is present at vertex v, occurs at v exactly r times. The r-maximum index [...] χr′(G) $\chi _r^\prime (G)$ is defined to be the minimum number k of colors needed for an r-maximum k-edge-coloring of graph G. In this paper we show that [...] χr′(G)≤3 $\chi _r^\prime (G) \le 3$ for any nontrivial connected graph G and r = 1 or 2. The bound 3 is tight. All graphs G with [...] χ1′(G)=i $\chi _1^\prime (G) = i$ , i = 1, 2, 3 are characterized. The precise value of the r-maximum index, r ≥ 1, is determined for trees and complete graphs.