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Rainbow Connectivity of Cacti and of Some Infinite Digraphs

100%
Alva-Samos J., Montellano-Ballesteros J. J.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 2
301-313
EN
An arc-coloured digraph D = (V,A) is said to be rainbow connected if for every pair {u, v} ⊆ V there is a directed uv-path all whose arcs have different colours and a directed vu-path all whose arcs have different colours. The minimum number of colours required to make the digraph D rainbow connected is called the rainbow connection number of D, denoted rc⃗ (D). A cactus is a digraph where each arc belongs to exactly one directed cycle. In this paper we give sharp upper and lower bounds for the rainbow connection number of a cactus and characterize those cacti whose rainbow connection number is equal to any of those bounds. Also, we calculate the rainbow con- nection numbers of some infinite digraphs and graphs, and present, for each n ≥ 6, a tournament of order n and rainbow connection number equal to 2.
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Bounds for the rainbow connection number of graphs

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Schiermeyer I.
Discussiones Mathematicae Graph Theory
|
2011
|
tom 31
|
nr 2
387-395
EN
An edge-coloured graph G is rainbow-connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow-connected. In this paper we show some new bounds for the rainbow connection number of graphs depending on the minimum degree and other graph parameters. Moreover, we discuss sharpness of some of these bounds.
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