Some Results on the Rainbow Vertex-Disconnection Colorings of Graphs.

Let G be a nontrivial connected and vertex-colored graph. A vertex subset X is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of G - S ; whereas when x and y are adjacent, S + x or S + y is rainbow and x and y belong to different components of (G - x y) - S . For a connected graph G, the rainbow vertex-disconnection number of G, rvd(G), is the minimum number of colors that are needed to make G rainbow vertex-disconnected. In this paper, we prove for any K 4 -minor free graph, r v d (G) ≤ Δ (G) and the bound is sharp. We show it is NP-complete to determine the rainbow vertex-disconnection numbers for bipartite graphs and split graphs. Moreover, we show for every ϵ > 0 , it is impossible to efficiently approximate the rainbow vertex-disconnection number of any bipartite graph and split graph within a factor of n 1 3 - ϵ unless Z P P = N P . [ABSTRACT FROM AUTHOR]