Article

Upper Bounds for the Rainbow Connection Numbers of Line Graphs

Graphs and Combinatorics (Impact Factor: 0.39). 01/2010; 28(2). 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]. "
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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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    • "[5] [6], "
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    ABSTRACT: A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. For any two vertices u and v of G, a rainbow u-v geodesic in G is a rainbow u-v path of length d(u,v), where d(u,v) is the distance between u and v. The graph G is strongly rainbow connected if there exists a rainbow u-v geodesic for any two vertices u and v in G. The strong rainbow connection number of G, denoted by src(G), is the minimum number of colors that are needed in order to make G strongly rainbow connected. In this paper, we first give a sharp upper bound for src(G) in terms of the number of edge-disjoint triangles in a graph G, and give a necessary and sufficient condition for the equality. We next investigate the graphs with large strong rainbow connection numbers. Chartrand et al. obtained that src(G)=m if and only if G is a tree, we will show that src(G)=m-1, and characterize the graphs G with src(G)=m-2 where m is the number of edges of G.
    Full-text · Article · Jan 2013 · The Bulletin of the Malaysian Mathematical Society Series 2
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    • "Theorem 2.37 [43] "
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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
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