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    Rainbow Hamilton cycles in random regular graphs

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    A rainbow subgraph of an edge-coloured graph has all edges of distinct colours. A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n tends to infinity, provided d is at least 8.Comment: 16 page

    Rainbow Connection of Random Regular Graphs

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    An edge colored graph GG is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph GG, denoted by rc(G)rc(G), is the smallest number of colors that are needed in order to make GG rainbow connected. In this work we study the rainbow connection of the random rr-regular graph G=G(n,r)G=G(n,r) of order nn, where r4r\ge 4 is a constant. We prove that with probability tending to one as nn goes to infinity the rainbow connection of GG satisfies rc(G)=O(logn)rc(G)=O(\log n), which is best possible up to a hidden constant

    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

    Hardness and Algorithms for Rainbow Connectivity

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    An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In addition to being a natural combinatorial problem, the rainbow connectivity problem is motivated by applications in cellular networks. In this paper we give the first proof that computing rc(G) is NP-Hard. In fact, we prove that it is already NP-Complete to decide if rc(G) = 2, and also that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every ϵ\epsilon > 0, a connected graph with minimum degree at least ϵn\epsilon n has bounded rainbow connectivity, where the bound depends only on ϵ\epsilon, and the corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also pre sented
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