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    Rainbow Connection Number and Connected Dominating Sets

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    Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by {\gamma}_c(G) + 2, where {\gamma}_c(G) is the connected domination number of G. Bounds of the form diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and chain graphs all with minimum degree at least 2 and connected. We also show that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of these cases, we also demonstrate the tightness of the bounds. An extension of this idea to two-step dominating sets is used to show that for every connected graph on n vertices with minimum degree {\delta}, the rainbow connection number is upper bounded by 3n/({\delta} + 1) + 3. This solves an open problem of Schiermeyer (2009), improving the previously best known bound of 20n/{\delta} by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up to additive factors by a construction of Caro et al. (2008).Comment: 14 page

    Note on the upper bound of the rainbow index of a graph

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    A path in an edge-colored graph GG, where adjacent edges may be colored the same, is a rainbow path if every two edges of it receive distinct colors. The rainbow connection number of a connected graph GG, denoted by rc(G)rc(G), is the minimum number of colors that are needed to color the edges of GG such that there exists a rainbow path connecting every two vertices of GG. Similarly, a tree in GG is a rainbow~tree if no two edges of it receive the same color. The minimum number of colors that are needed in an edge-coloring of GG such that there is a rainbow tree connecting SS for each kk-subset SS of V(G)V(G) is called the kk-rainbow index of GG, denoted by rxk(G)rx_k(G), where kk is an integer such that 2kn2\leq k\leq n. Chakraborty et al. got the following result: For every ϵ>0\epsilon> 0, a connected graph with minimum degree at least ϵn\epsilon n has bounded rainbow connection, where the bound depends only on ϵ\epsilon. Krivelevich and Yuster proved that if GG has nn vertices and the minimum degree δ(G)\delta(G) then rc(G)<20n/δ(G)rc(G)<20n/\delta(G). This bound was later improved to 3n/(δ(G)+1)+33n/(\delta(G)+1)+3 by Chandran et al. Since rc(G)=rx2(G)rc(G)=rx_2(G), a natural problem arises: for a general kk determining the true behavior of rxk(G)rx_k(G) as a function of the minimum degree δ(G)\delta(G). In this paper, we give upper bounds of rxk(G)rx_k(G) in terms of the minimum degree δ(G)\delta(G) in different ways, namely, via Szemer\'{e}di's Regularity Lemma, connected 22-step dominating sets, connected (k1)(k-1)-dominating sets and kk-dominating sets of GG.Comment: 12 pages. arXiv admin note: text overlap with arXiv:0902.1255 by other author
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