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# Upper bounds for the rainbow connection numbers of line graphs

• ##### Yuefang Sun
Graphs and Combinatorics (Impact Factor: 0.35). 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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##### 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 $G$.
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##### Article: Rainbow Colouring of Split and Threshold Graphs
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ABSTRACT: A rainbow colouring of a connected graph is a colouring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are coloured the same. Such a colouring using minimum possible number of colours is called an optimal rainbow colouring, and the minimum number of colours required is called the rainbow connection number of the graph. In this article, we show the following: 1. The problem of deciding whether a graph can be rainbow coloured using 3 colours remains NP-complete even when restricted to the class of split graphs. However, any split graph can be rainbow coloured in linear time using at most one more colour than the optimum. 2. For every integer k larger than 2, the problem of deciding whether a graph can be rainbow coloured using k colours remains NP-complete even when restricted to the class of chordal graphs. 3. For every positive integer k, threshold graphs with rainbow connection number k can be characterised based on their degree sequence alone. Further, we can optimally rainbow colour a threshold graph in linear time.
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##### 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. · 0.35 Impact Factor