Geodesic
Maximum Edge-Colorings Of Graphs
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 1, pp. 117-125

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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 d_G(v) = d, d ≥ r, the maximum color, that is present at vertex v, occurs at v exactly r times. The r-maximum index χ_r^′ (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 χ_r^′ (G) ≤ 3 for any nontrivial connected graph G and r = 1 or 2. The bound 3 is tight. All graphs G with χ_1^' (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.
Keywords: edge-coloring, r -maximum k -edge-coloring, unique-maximum edge-coloring, weak-odd edge-coloring, weak-even edge-coloring 
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Jendrol’, Stanislav; Vrbjarová, Michaela. Maximum Edge-Colorings Of Graphs. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 1, pp. 117-125. http://geodesic.mathdoc.fr/item/DMGT_2016_36_1_a8/