Rainbow connection in graphs

Gary Chartrand; Garry L. Johns; Kathleen A. McKeon; Ping Zhang

Mathematica Bohemica (2008)

  • Volume: 133, Issue: 1, page 85-98
  • ISSN: 0862-7959

Abstract

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Let G be a nontrivial connected graph on which is defined a coloring c E ( G ) { 1 , 2 , ... , k } , k , of the edges of G , where adjacent edges may be colored the same. A path P in G is a rainbow path if no two edges of P are colored the same. The graph G is rainbow-connected if G contains a rainbow u - v path for every two vertices u and v of G . The minimum k for which there exists such a k -edge coloring is the rainbow connection number r c ( G ) of G . If for every pair u , v of distinct vertices, G contains a rainbow u - v geodesic, then G is strongly rainbow-connected. The minimum k for which there exists a k -edge coloring of G that results in a strongly rainbow-connected graph is called the strong rainbow connection number s r c ( G ) of G . Thus r c ( G ) s r c ( G ) for every nontrivial connected graph G . Both r c ( G ) and s r c ( G ) are determined for all complete multipartite graphs G as well as other classes of graphs. For every pair a , b of integers with a 3 and b ( 5 a - 6 ) / 3 , it is shown that there exists a connected graph G such that r c ( G ) = a and s r c ( G ) = b .

How to cite

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Chartrand, Gary, et al. "Rainbow connection in graphs." Mathematica Bohemica 133.1 (2008): 85-98. <http://eudml.org/doc/32579>.

@article{Chartrand2008,
abstract = {Let $G$ be a nontrivial connected graph on which is defined a coloring $c\: E(G) \rightarrow \lbrace 1, 2, \ldots , k\rbrace $, $k \in \{\mathbb \{N\}\}$, of the edges of $G$, where adjacent edges may be colored the same. A path $P$ in $G$ is a rainbow path if no two edges of $P$ are colored the same. The graph $G$ is rainbow-connected if $G$ contains a rainbow $u-v$ path for every two vertices $u$ and $v$ of $G$. The minimum $k$ for which there exists such a $k$-edge coloring is the rainbow connection number $\mathop \{\mathrm \{r\}c\}(G)$ of $G$. If for every pair $u, v$ of distinct vertices, $G$ contains a rainbow $u-v$ geodesic, then $G$ is strongly rainbow-connected. The minimum $k$ for which there exists a $k$-edge coloring of $G$ that results in a strongly rainbow-connected graph is called the strong rainbow connection number $\mathop \{\mathrm \{s\}rc\}(G)$ of $G$. Thus $\mathop \{\mathrm \{r\}c\}(G) \le \mathop \{\mathrm \{s\}rc\}(G)$ for every nontrivial connected graph $G$. Both $\mathop \{\mathrm \{r\}c\}(G)$ and $\mathop \{\mathrm \{s\}rc\}(G)$ are determined for all complete multipartite graphs $G$ as well as other classes of graphs. For every pair $a, b$ of integers with $a \ge 3$ and $b \ge (5a-6)/3$, it is shown that there exists a connected graph $G$ such that $\mathop \{\mathrm \{r\}c\}(G)=a$ and $\mathop \{\mathrm \{s\}rc\}(G)=b$.},
author = {Chartrand, Gary, Johns, Garry L., McKeon, Kathleen A., Zhang, Ping},
journal = {Mathematica Bohemica},
keywords = {edge coloring; rainbow coloring; strong rainbow coloring; edge coloring; rainbow path; rainbow coloring; strong rainbow coloring; rainbow connection number; strong rainbow connection number},
language = {eng},
number = {1},
pages = {85-98},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Rainbow connection in graphs},
url = {http://eudml.org/doc/32579},
volume = {133},
year = {2008},
}

TY - JOUR
AU - Chartrand, Gary
AU - Johns, Garry L.
AU - McKeon, Kathleen A.
AU - Zhang, Ping
TI - Rainbow connection in graphs
JO - Mathematica Bohemica
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 133
IS - 1
SP - 85
EP - 98
AB - Let $G$ be a nontrivial connected graph on which is defined a coloring $c\: E(G) \rightarrow \lbrace 1, 2, \ldots , k\rbrace $, $k \in {\mathbb {N}}$, of the edges of $G$, where adjacent edges may be colored the same. A path $P$ in $G$ is a rainbow path if no two edges of $P$ are colored the same. The graph $G$ is rainbow-connected if $G$ contains a rainbow $u-v$ path for every two vertices $u$ and $v$ of $G$. The minimum $k$ for which there exists such a $k$-edge coloring is the rainbow connection number $\mathop {\mathrm {r}c}(G)$ of $G$. If for every pair $u, v$ of distinct vertices, $G$ contains a rainbow $u-v$ geodesic, then $G$ is strongly rainbow-connected. The minimum $k$ for which there exists a $k$-edge coloring of $G$ that results in a strongly rainbow-connected graph is called the strong rainbow connection number $\mathop {\mathrm {s}rc}(G)$ of $G$. Thus $\mathop {\mathrm {r}c}(G) \le \mathop {\mathrm {s}rc}(G)$ for every nontrivial connected graph $G$. Both $\mathop {\mathrm {r}c}(G)$ and $\mathop {\mathrm {s}rc}(G)$ are determined for all complete multipartite graphs $G$ as well as other classes of graphs. For every pair $a, b$ of integers with $a \ge 3$ and $b \ge (5a-6)/3$, it is shown that there exists a connected graph $G$ such that $\mathop {\mathrm {r}c}(G)=a$ and $\mathop {\mathrm {s}rc}(G)=b$.
LA - eng
KW - edge coloring; rainbow coloring; strong rainbow coloring; edge coloring; rainbow path; rainbow coloring; strong rainbow coloring; rainbow connection number; strong rainbow connection number
UR - http://eudml.org/doc/32579
ER -

References

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  1. Introduction to Graph Theory, McGraw-Hill, Boston, 2005. (2005) 

Citations in EuDML Documents

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  1. Futaba Fujie-Okamoto, Kyle Kolasinski, Jianwei Lin, Ping Zhang, Vertex rainbow colorings of graphs
  2. Xueliang Li, Yongtang Shi, On the Rainbow Vertex-Connection
  3. Lily Chen, Xueliang Li, Kang Yang, Yan Zhao, The 3-Rainbow Index of a Graph
  4. Xueliang Li, Mengmeng Liu, Ingo Schiermeyer, Rainbow Connection Number of Dense Graphs
  5. Arnfried Kemnitz, Ingo Schiermeyer, Graphs with rainbow connection number two
  6. Arnfried Kemnitz, Jakub Przybyło, Ingo Schiermeyer, Mariusz Woźniak, Rainbow Connection In Sparse Graphs
  7. Xueliang Li, Ingo Schiermeyer, Kang Yang, Yan Zhao, Graphs with 3-Rainbow Index n − 1 and n − 2
  8. Ingo Schiermeyer, Bounds for the rainbow connection number of graphs
  9. Xueliang Li, Ingo Schiermeyer, Kang Yang, Yan Zhao, Graphs with 4-Rainbow Index 3 and n − 1
  10. Sandi Klavžar, Gašper Mekiš, On the rainbow connection of Cartesian products and their subgraphs

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