Geodesic
M2-Edge Colorings Of Cacti And Graph Joins
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 1, pp. 59-69

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An edge coloring φ of a graph G is called an M2-edge coloring if |φ(v)| ≤ 2 for every vertex v of G, where φ(v) is the set of colors of edges incident with v. Let K2(G) denote the maximum number of colors used in an M2-edge coloring of G. In this paper we determine K2(G) for trees, cacti, complete multipartite graphs and graph joins.
Keywords: cactus, edge coloring, graph join 
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Czap, Július; Šugerek, Peter; Ivančo, Jaroslav. M2-Edge Colorings Of Cacti And Graph Joins. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 1, pp. 59-69. http://geodesic.mathdoc.fr/item/DMGT_2016_36_1_a4/