Further Results on the Rainbow Vertex-Disconnection of Graphs

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set is called rainbow if any two vertices in have distinct colors. The graph vertex-disconnected for x y G, there exists S such that when are nonadjacent, belong to different components $$G-S$$ ; whereas adjacent, $$S+x$$ or $$S+y$$ $$(G-xy)-S$$ . Such an x–y vertex-cut G. For vertex-disconnection number denoted by rvd(G), minimum colors needed make vertex-disconnected. In this paper, we obtain bounds terms degree maximum We give tighter upper bound size with $$rvd(G)=k$$ $$k\ge \frac{n}{2}$$ then characterize graphs order n $$n-1$$ $$rvd(G)=n-1$$ Moreover, get sharp threshold function property $$rvd(G(n,p))=n$$ prove almost all $$rvd(G)=rvd({\overline{G}})=n$$ Finally, some Nordhaus–Gaddum-type results: $$n-5\le rvd(G)+rvd({\overline{G}})\le 2n$$ $$n-1\le rvd(G)\cdot rvd({\overline{G}})\le n^2$$ numbers $${\overline{G}}$$ $$n\ge 24$$