Some results on the 3-vertex-rainbow index of a graph
type: Item
instance of: scholarly article
title: Some results on the 3-vertex-rainbow index of a graph
zbMATH Open document ID: 1475.05071
author: Yingbin Ma
author: Wenhan Zhu
published in: Bulletin of the Malaysian Mathematical Sciences Society. Second Series
publication date: 2021-04-27T00:00:00Z
review text: In this paper, the authors investigate the 3-vertex-rainbow index \(rvx_3\) of unicyclic graphs and complementary graphs, respectively. Initially, they calculate 3-vertex-rainbow index \(rvx_3\) of a cycle \(C_n\). Theorem: For an integer \(n\), \[ r v x_{3}\left(C_{n}\right)= \begin{cases}0 & \text { if } \quad 3 \leq n \leq 4; \\ n-4 & \text { if } \quad n \geq 5.\end{cases} \] In the next result, it is proved that if \(G\) is a unicyclic graph of order \(n \geq 4\) that is not a cycle and the girth \(g(G) \in \{3,4\}\), then \(rvx_3(G) \leq n - 3\), and further it is proved that this bound is tight. Similarly, two more results are proved dealing with unicyclic graphs having girth 5 and girth at least 6. In fact, the following results are proved. Theorem: If \(G\) is a unicyclic graph of order \(n \geq 6\) that is not a cycle and girth \(5\), then \(rvx_3(G) \leq n - 4\), and the bound is tight. Theorem: If \(G\) is a unicyclic graph of order \(n \geq 7\) that is not a cycle and girth at least \(6\), then \(rvx_3(G) \leq n - 5\), and the bound is tight. Lastly, they investigate the 3-vertex-rainbow index of a connected graph \(G\) using the complement \(\bar{G}\) of \(G\) with diameter at least 3. Theorem: Let \(G\) be a connected graph of order \(n\). \par (i) If \(\bar{G}\) is connected and diam \((\bar{G}) \geq 4\), then \(r v x_{3}(G)=1\). \par (ii) If \(\bar{G}\) is connected and \(\operatorname{diam}(\bar{G})=3\), then \(1 \leq r v x_{3}(G) \leq 2\), and the bounds are tight. \par (iii) If \(\bar{G}\) is disconnected, then \(0 \leq r v x_{3}(G) \leq 1\), and the bounds are tight.
reviewed by: Vinayak Joshi
Mathematics Subject Classification ID: 05C15
Mathematics Subject Classification ID: 05C40
Mathematics Subject Classification ID: 05C09
zbMATH DE Number: 7340053
zbMATH Keywords: vertex-coloring
zbMATH Keywords: \(S\)-tree
zbMATH Keywords: vertex-rainbow \(S\)-tree
zbMATH Keywords: \(k\)-vertex-rainbow index
MaRDI profile type: Publication
full work available at URL: s40840-020-00992-x
OpenAlex ID: W3048909974
cites work: Q5422499
cites work: Rainbow connection number and connected dominating sets
cites work: Rainbow connection in graphs
cites work: Rainbow trees in graphs and generalized connectivity
cites work: The 3-rainbow index of a graph
cites work: Rainbow connections for outerplanar graphs with diameter 2 and 3
cites work: The rainbow connection of a graph is (at most) reciprocal to its minimum degree
cites work: Rainbow vertex connection of digraphs
cites work: Rainbow connection number and graph operations
cites work: Rainbow connection number and connectivity
cites work: Graphs with 3-rainbow index \(n-1\) and \(n-2\)
cites work: On the rainbow vertex-connection
cites work: Rainbow connections of graphs: a survey
cites work: Rainbow vertex \(k\)-connection in graphs
cites work: Total rainbow \(k\)-connection in graphs
cites work: Total rainbow connection number and complementary graph
cites work: Rainbow connection numbers of Cayley graphs
cites work: Rainbow connection numbers of Cayley digraphs on abelian groups
cites work: The vertex-rainbow index of a graph
cites work: On minimally rainbow \(k\)-connected graphs
DOI: 10.1007/S40840-020-00992-X
Recommended article: Some upper bounds for 3-rainbow index of graphs
Recommended article: The 3-rainbow index of a graph
Recommended article: Q4977513
Recommended article: Graphs with 3-rainbow index \(n-1\) and \(n-2\)
Recommended article: The strong 3-rainbow index of some certain graphs and its amalgamation
Recommended article: Some results on the 3-total-rainbow index
Recommended article: The 3-rainbow index and connected dominating sets
Recommended article: Q3297141
Recommended article: The strong 3-rainbow index of edge-amalgamation of some graphs
Recommended article: Note on the vertex-rainbow index of a graph
description: scientific article; zbMATH DE number 7340053