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 \((G-xy)-S\). 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, \(rvd(G)\le \Delta (G)\) 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 \(\epsilon >0\), it is impossible to efficiently approximate the rainbow vertex-disconnection number of any bipartite graph and split graph within a factor of \(n^{\frac{1}{3}-\epsilon }\) unless \(ZPP=NP\).