Geodesic
Vertex-distinguishing edge-colorings of linear forests
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 1, pp. 95-103

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In the PhD thesis by Burris (Memphis (1993)), a conjecture was made concerning the number of colors c(G) required to edge-color a simple graph G so that no two distinct vertices are incident to the same multiset of colors. We find the exact value of c(G) - the irregular coloring number, and hence verify the conjecture when G is a vertex-disjoint union of paths. We also investigate the point-distinguishing chromatic index, χ₀(G), where sets, instead of multisets, are required to be distinct, and determine its value for the same family of graphs.
Keywords: irregular edge-coloring, vertex-distinguishing edge-coloring, point-distinguishing chromatic index 
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Cichacz, Sylwia; Przybyło, Jakub. Vertex-distinguishing edge-colorings of linear forests. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 1, pp. 95-103. http://geodesic.mathdoc.fr/item/DMGT_2010_30_1_a7/