Let G be a nontrivial edge-colored connected graph. An edge-cut R of G is called a rainbow cut if no two edges of it are colored the same. An edge-colored graph G is rainbow disconnected if for every two vertices u and v, there exists a u − v rainbow cut. 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 first solve a conjecture that determines the maximum size of a connected graph G of order n with rd(G) = k for given integers k and n with 1 ≤ k ≤ n − 1, where n is odd, posed by Chartrand et al. in [5]. Secondly, we discuss bounds of the rainbow disconnection numbers for complete multipartite graphs, critical graphs, minimal graphs with respect to chromatic index and regular graphs, and give the rainbow disconnection numbers for several special graphs. Finally, we get the Nordhaus-Gaddum-type theorem for the rainbow disconnection number of graphs. We prove that if G and G are both connected, then nFurthermore, examples are given to show that the upper bounds are sharp for n ≥ 6, and the lower bounds are sharp when G = G = P 4 .

Let G be a nontrivial connected and vertex colored graph. A vertex cut S of G is called a rainbow vertex cut if no two vertices of it are colored the same. The graph G is called rainbow vertex disconnected if for any two nonadjacent vertices x and y, there exists an x − y rainbow vertex cut. We introduce and study the rainbow vertex disconnection number rvd(G) of a connected graph G, which is defined as the minimum number of colors that are needed to make G rainbow vertex disconnected. In this paper, we first characterize the connected graphs G with rvd(G) = 1 and n, respectively. We also characterize the minimally 2connected graphs G for which rvd(G) = 2 and n − 2, respectively. Secondly, we determine the rainbow vertex disconnection numbers for complete multipartite graphs and grid graphs. Finally, we get the maximum size of a connected graph G of order n with rvd(G) = k for given integers k and n with 1 ≤ k ≤ n.

For an edge-colored graph G, a set F of edges of G is called a proper cut if F is an edge-cut of G and any pair of adjacent edges in F are assigned by different colors. An edge-colored graph is proper disconnected if for each pair of distinct vertices of G there exists a proper edge-cut separating them. For a connected graph G, the proper disconnection number of G, denoted by pd(G), is the minimum number of colors that are needed in order to make G proper disconnected. In this paper, we first give the exact values of the proper disconnection numbers for some special families of graphs. Next, we obtain a sharp upper bound of pd(G) for a connected graph G of order n, i.e, pd(G) ≤ min{χ ′ (G) − 1, n 2 }. Finally, we show that for given integers k and n, the minimum size of a connected graph G of order n with pd(G) = k is n − 1 for k = 1 and n + 2k − 4 for 2 ≤ k ≤ ⌈ n 2 ⌉.

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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.