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Let $G$ be a nontrivial connected and vertex-colored graph. A subset $X$ of the vertex set of $G$ is called rainbow if any two vertices in $X$ have distinct colors. The graph $G$ is called emph{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 $(G-xy)-S$. Such a vertex subset $S$ is called a emph{rainbow vertex-cut} of $G$. For a connected graph $G$, the emph{rainbow vertex-disconnection number} of $G$, denoted by $rvd(G)$, is the minimum number of colors that are needed to make $G$ rainbow vertex-disconnected. In this paper, we obtain bounds of the rainbow vertex-disconnection number of a graph in terms of the minimum degree and maximum degree of the graph. We give a tighter upper bound for the maximum size of a graph $G$ with $rvd(G)=k$ for $kgeqfrac{n}{2}$. We then characterize the graphs of order $n$ with rainbow vertex-disconnection number $n-1$ and obtain the maximum size of a graph $G$ with $rvd(G)=n-1$. Moreover, we get a sharp threshold function for the property $rvd(G(n,p))=n$ and prove that almost all graphs $G$ have $rvd(G)=rvd(overline{G})=n$. Finally, we obtain some Nordhaus-Gaddum-type results: $n-5leq rvd(G)+rvd(overline{G})leq 2n$ and $n-1leq rvd(G)cdot rvd(overline{G})leq n^2$ for the rainbow vertex-disconnection numbers of nontrivial connected graphs $G$ and $overline{G}$ with order $ngeq 24$.
Let $G$ be a nontrivial edge-colored connected graph. An edge-cut $R$ of $G$ is called a {it rainbow edge-cut} if no two edges of $R$ are colored with the same color. For two distinct vertices $u$ and $v$ of $G$, if an edge-cut separates them, then t
An edge-cut $R$ of an edge-colored connected graph is called a rainbow-cut if no two edges in the edge-cut are colored the same. An edge-colored graph is rainbow disconnected if for any two distinct vertices $u$ and $v$ of the graph, there exists a $
A emph{proper $t$-edge-coloring} of a graph $G$ is a mapping $alpha: E(G)rightarrow {1,ldots,t}$ such that all colors are used, and $alpha(e) eq alpha(e^{prime})$ for every pair of adjacent edges $e,e^{prime}in E(G)$. If $alpha $ is a proper edge-col
We provide precise asymptotic estimates for the number of several classes of labelled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky et al. (Random Struct
A close relation between hitting times of the simple random walk on a graph, the Kirchhoff index, resistance-centrality, and related invariants of unicyclic graphs is displayed. Combining with the graph transformations and some other techniques, shar