Rainbow vertex-connection and forbidden subgraphs
Wenjing Li, Xueliang Li, Jingshu Zhang

TL;DR
This paper characterizes specific graph families defined by forbidden subgraphs for which the rainbow vertex-connection number is tightly bounded by the graph's diameter plus a constant.
Contribution
It identifies all connected graph families with one or two forbidden subgraphs where the rainbow vertex-connection number is linearly bounded by diameter plus a constant.
Findings
Characterizes families with bounded rainbow vertex-connection number
Establishes bounds related to graph diameter for these families
Provides a complete classification for one- and two-forbidden subgraph cases
Abstract
A path in a vertex-colored graph is called \emph{vertex-rainbow} if its internal vertices have pairwise distinct colors. A graph is \emph{rainbow vertex-connected} if for any two distinct vertices of , there is a vertex-rainbow path connecting them. For a connected graph , the \emph{rainbow vertex-connection number} of , denoted by , is defined as the minimum number of colors that are required to make rainbow vertex-connected. In this paper, we find all the families of connected graphs with , for which there is a constant such that, for every connected -free graph , , where is the diameter of .
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Graph Labeling and Dimension Problems
