Geodesic
Rainbow Disconnection in Graphs
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 4, pp. 1007-1021.

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Let G be a nontrivial connected, edge-colored graph. An edge-cut R of G is called a rainbow cut if no two edges in R are colored the same. An edge-coloring of G is a rainbow disconnection coloring if for every two distinct vertices u and v of G, there exists a rainbow cut in G, where u and v belong to different components of G − R. We introduce and study the rainbow disconnection number rd(G) of G, which is defined as the minimum number of colors required of a rainbow disconnection coloring of G. It is shown that the rainbow disconnection number of a nontrivial connected graph G equals the maximum rainbow disconnection number among the blocks of G. It is also shown that for a nontrivial connected graph G of order n, rd(G) = n−1 if and only if G contains at least two vertices of degree n − 1. The rainbow disconnection numbers of all grids Pm □ Pn are determined. Furthermore, it is shown for integers k and n with 1 ≤ k ≤ n − 1 that the minimum size of a connected graph of order n having rainbow disconnection number k is n + k − 2. Other results and a conjecture are also presented.
Keywords: edge coloring, rainbow connection, rainbow disconnection 
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Chartrand, Gary; Devereaux, Stephen; Haynes, Teresa W.; Hedetniemi, Stephen T.; Zhang, Ping. Rainbow Disconnection in Graphs. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 4, pp. 1007-1021. http://geodesic.mathdoc.fr/item/DMGT_2018_38_4_a9/

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