Article

Oriented diameter and rainbow connection number of a graph

Nankai University, T’ien-ching-shih, Tianjin Shi, China
Discrete mathematics & theoretical computer science DMTCS (Impact Factor: 0.32). 11/2011; 16(3).
Source: arXiv

ABSTRACT

The oriented diameter of a bridgeless graph $G$ is $\min\{diam(H)\ | H\ is\
an orientation\ of\ G\}$. A path in an edge-colored graph $G$, where adjacent
edges may have the same color, is called rainbow if no two edges of the path
are colored the same. The rainbow connection number $rc(G)$ of $G$ is the
smallest integer $k$ for which there exists a $k$-edge-coloring of $G$ such
that every two distinct vertices of $G$ are connected by a rainbow path. In
this paper, we obtain upper bounds for the oriented diameter and the rainbow
connection number of a graph in terms of $rad(G)$ and $\eta(G)$, where $rad(G)$
is the radius of $G$ and $\eta(G)$ is the smallest integer number such that
every edge of $G$ is contained in a cycle of length at most $\eta(G)$. We also
obtain constant bounds of the oriented diameter and the rainbow connection
number for a (bipartite) graph $G$ in terms of the minimum degree of $G$.

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Available from: Xueliang Li, Jan 12, 2014
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    ABSTRACT: The concept of rainbow connection was introduced by Chartrand et al. in 2008. It is fairly interesting and recently quite a lot papers have been published about it. In this survey we attempt to bring together most of the results and papers that dealt with it. We begin with an introduction, and then try to organize the work into five categories, including (strong) rainbow connection number, rainbow $k$-connectivity, $k$-rainbow index, rainbow vertex-connection number, algorithms and computational complexity. This survey also contains some conjectures, open problems or questions.
    Full-text · Article · Jan 2013 · Graphs and Combinatorics