Article

# Upper bounds for the rainbow connection numbers of line graphs

(Impact Factor: 0.33). 01/2010; DOI: 10.1007/s00373-011-1034-1
Source: arXiv

ABSTRACT A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of it are colored the same. A nontrivial connected graph $G$ is rainbow connected if for any two vertices of $G$ there is a rainbow path connecting them. The rainbow connection number of $G$, denoted by $rc(G)$, is defined as the smallest number of colors by using which there is a coloring such that $G$ is rainbow connected. In this paper, we mainly study the rainbow connection number of the line graph of a graph which contains triangles and get two sharp upper bounds for $rc(L(G))$, in terms of the number of edge-disjoint triangles of $G$ where $L(G)$ is the line graph of $G$. We also give results on the iterated line graphs. Comment: 11 pages

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Available from: Xueliang Li, Feb 03, 2014
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• "Actually, it has been proved in [23], that for any fixed t ≥ 2, deciding if rc(Γ) = t is NP-complete. Some topics on restrict graphs are as follows: oriented graphs [8], graph products [15], hypergraphs [4], corona graphs [9], line graphs [21], Cayley graphs [22], dense graphs [20] and sparse random graphs [12]. Most of the results and papers that dealt with it can be found in [19]. "
##### Article: The rainbow connection number of the power graph of a finite group
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ABSTRACT: This paper studies the rainbow connection number of the power graph $\Gamma_G$ of a finite group $G$. We determine the rainbow connection number of $\Gamma_G$ if $G$ has maximal involutions or is nilpotent, and show that the rainbow connection number of $\Gamma_G$ is at most three if $G$ has no maximal involutions. The rainbow connection numbers of power graphs of some nonnilpotent groups are also given.
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• "Theorem 2.37 [43] "
##### Article: Rainbow Connections of Graphs: A Survey
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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.
Graphs and Combinatorics 01/2013; 29(1):1--38. DOI:10.1007/s00373-012-1243-2 · 0.33 Impact Factor
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• "Bounds for the rainbow connection number of a graph have also been studies in terms of other graph parameters, for example, radius, dominating number, minimum degree, connectivity, etc. [1] [4] [10]. Cayley graphs and line graphs were studied in [12] and [13], respectively. "
##### Article: Oriented diameter and rainbow connection number of a graph
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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 $G$.
Discrete mathematics & theoretical computer science DMTCS 11/2011; · 0.61 Impact Factor