The strong rainbow vertex-connection of graphs

TL;DR

This paper investigates the strong rainbow vertex-connection number of graphs, establishing bounds, characterizations for specific values, and demonstrating the existence of graphs with prescribed rainbow connection parameters.

Contribution

It provides sharp bounds for the strong rainbow vertex-connection number and characterizes graphs achieving extremal values, also showing the existence of graphs with specified rainbow connection numbers.

Findings

Bounds for $srvc(G)$ are between 0 and $n-2$.

Graphs with $srvc(G)=1, 2, n-2$ are characterized.

Existence of graphs with given $rvc(G)$ and $srvc(G)$ values is proven.

Abstract

A vertex-colored graph $G$ is said to be rainbow vertex-connected if every two vertices of $G$ are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. If for every pair $u, v$ of distinct vertices, $G$ contains a rainbow $u-v$ geodesic, then $G$ is strong rainbow vertex-connected. The minimum number $k$ for which there exists a $k$-vertex-coloring of $G$ that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of $G$, denoted by $srvc(G)$. Observe that $rvc(G)\leq srvc(G)$ for any nontrivial connected graph $G$. In this paper, sharp upper and lower bounds of $srvc(G)$ are given for a connected graph $G$ of order $n$, that is, $0\leq srvc(G)\leq n-2$. Graphs of order $n$ such that $srvc(G)= 1, 2, n-2$ are characterized, respectively. It is also shown that, for each pair $a, b$ of integers with $a\geq 5$ and $b\geq (7a-8)/5$, there exists a connected graph $G$ such that $rvc(G)=a$ and $srvc(G)=b$.