Geodesic
Distance-Local Rainbow Connection Number
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 4, pp. 1027-1039

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Under an edge coloring (not necessarily proper), a rainbow path is a path whose edge colors are all distinct. The d-local rainbow connection number lrcd(G) (respectively, d-local strong rainbow connection number lsrcd(G)) is the smallest number of colors needed to color the edges of G such that any two vertices with distance at most d can be connected by a rainbow path (respectively, rainbow geodesic). This generalizes rainbow connection numbers, which are the special case d = diam(G). We discuss some bounds and exact values. Moreover, we also characterize all triples of positive integers d, a, b such that there is a connected graph G with lrcd(G) = a and lsrcd(G) = b.
Keywords: rainbow connection, chromatic number, line graph 
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Septyanto, Fendy; Sugeng, Kiki A. Distance-Local Rainbow Connection Number. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 4, pp. 1027-1039. http://geodesic.mathdoc.fr/item/DMGT_2022_42_4_a0/