#### Title

Rainbow Disconnection in Graphs

#### Document Type

Article

#### Publication Date

1-1-2018

#### Description

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.

#### Citation Information

Chartrand, Gary;
Devereaux, Stephen;
Haynes, Teresa W.;
Hedetniemi, Stephen T.;
and
Zhang, Ping.
2018.
Rainbow Disconnection in Graphs.
*Discussiones Mathematicae - Graph Theory*.
Vol.38(4).
1007-1021.
https://doi.org/10.7151/dmgt.2061
ISSN: 1234-3099