Further results on the rainbow vertex-disconnection of graphs
Xueliang Li, Yindi Weng

TL;DR
This paper investigates the rainbow vertex-disconnection number in graphs, providing bounds, characterizations, and probabilistic thresholds, and explores relationships between a graph and its complement.
Contribution
It introduces new bounds and characterizations for the rainbow vertex-disconnection number, including thresholds and Nordhaus-Gaddum-type results, advancing understanding of this graph invariant.
Findings
Bounds in terms of minimum and maximum degree
Characterization of graphs with rvd(G)=n-1
Sharp threshold for rvd(G(n,p))=n
Abstract
Let be a nontrivial connected and vertex-colored graph. A subset of the vertex set of is called rainbow if any two vertices in have distinct colors. The graph is called \emph{rainbow vertex-disconnected} if for any two vertices and of , there exists a vertex subset such that when and are nonadjacent, is rainbow and and belong to different components of ; whereas when and are adjacent, or is rainbow and and belong to different components of . Such a vertex subset is called a \emph{rainbow vertex-cut} of . For a connected graph , the \emph{rainbow vertex-disconnection number} of , denoted by , is the minimum number of colors that are needed to make rainbow vertex-disconnected. In this paper, we obtain bounds of the rainbow vertex-disconnection number of a graph in…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Graph Labeling and Dimension Problems
