Rainbow Connection of Random Regular Graphs

An edge colored graph $G$ 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 $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this work we study the rainbow connection of the random $r$-regular graph $G=G(n,r)$ of order $n$, where $r\ge 4$ is a constant. We prove that with probability tending to one as $n$ goes to infinity the rainbow connection of $G$ satisfies $rc(G)=O(\log n)$, which is best possible up to a hidden constant

Rainbow Hamilton cycles in random regular graphs

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

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

A path in an edge-colored graph $G$, 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 $G$, denoted by $rc(G)$, is the minimum number of colors that are needed to color the edges of $G$ such that there exists a rainbow path connecting every two vertices of $G$. Similarly, a tree in $G$ 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 $G$ such that there is a rainbow tree connecting $S$ for each $k$-subset $S$ of $V(G)$ is called the $k$-rainbow index of $G$, denoted by $rx_k(G)$, where $k$ is an integer such that $2\leq k\leq n$. Chakraborty et al. got the following result: For every $\epsilon> 0$, a connected graph with minimum degree at least $\epsilon n$ has bounded rainbow connection, where the bound depends only on $\epsilon$. Krivelevich and Yuster proved that if $G$ has $n$ vertices and the minimum degree $\delta(G)$ then $rc(G)<20n/\delta(G)$. This bound was later improved to $3n/(\delta(G)+1)+3$ by Chandran et al. Since $rc(G)=rx_2(G)$, a natural problem arises: for a general $k$ determining the true behavior of $rx_k(G)$ as a function of the minimum degree $\delta(G)$. In this paper, we give upper bounds of $rx_k(G)$ in terms of the minimum degree $\delta(G)$ in different ways, namely, via Szemer\'{e}di's Regularity Lemma, connected $2$-step dominating sets, connected $(k-1)$-dominating sets and $k$-dominating sets of $G$.Comment: 12 pages. arXiv admin note: text overlap with arXiv:0902.1255 by other author

Hardness and Algorithms for Rainbow Connectivity

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 $\epsilon$ > 0, a connected graph with minimum degree at least $\epsilon n$ has bounded rainbow connectivity, where the bound depends only on $\epsilon$, 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