Erdős-Gallai-type results for the rainbow disconnection number of graphs

Abstract

Let G be a nontrivial connected and edge-colored graph. An edge-cut R of G is called a rainbow cut if no two edges of it are colored with a same color. An edge-colored graph G is called rainbow disconnected if for every two distinct vertices u and v of G, there exists a u − v rainbow cut separating them. For a connected graph G, the rainbow disconnection number of G, denoted by rd(G), is defined as the smallest number of colors that are needed in order to make G rainbow disconnected. In this paper, we will study the Erdős-Gallai-type results for rd(G), and completely solve them.

Proper disconnection of graphs

“…An edge-coloring is called a rainbow disconnection coloring of G if for every two vertices of G, there exists a rainbow cut in G separating them. For a connected graph G, the rainbow disconnection number of G, denoted rd(G), is the smallest number of colors required for a rainbow disconnection coloring of G. A rainbow disconnection coloring with rd(G) colors is called an rd-coloring of G. In [1,2,8] the authors have obtained many results.…”

The size of graphs with restricted rainbow $2$-connection number

“…We remark that analogous Erdős-Gallai type problems have also been considered for other parameters similar to the rainbow connection number, such as the monochromatic connection number [5], and rainbow disconnection number [1], among others.…”