Mathematics – Combinatorics
Scientific paper
2012-01-22
Mathematics
Combinatorics
12 pages, 4 figures
Scientific paper
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 connectivity 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 connectivity of binomial random graphs at the connectivity threshold $p=\frac{\log n+\om}{n}$ where $\om=\om(n)\to\infty$ and ${\om}=o(\log{n})$ and of random $r$-regular graphs where $r \geq 3$ is a fixed integer. Specifically, we prove that the rainbow connectivity $rc(G)$ of $G=G(n,p)$ satisfies $rc(G) \sim \max\set{Z_1,diameter(G)}$ with high probability (\whp). Here $Z_1$ is the number of vertices in $G$ whose degree equals 1 and the diameter of $G$ is asymptotically equal to $\diam$ \whp. Finally, we prove that the rainbow connectivity $rc(G)$ of the random $r$-regular graph $G=G(n,r)$ satisfies $rc(G) =O(\log^2{n})$ \whp.
Frieze Alan
Tsourakakis Charalampos E.
No associations
LandOfFree
Rainbow Connectivity of Sparse Random Graphs does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Rainbow Connectivity of Sparse Random Graphs, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Rainbow Connectivity of Sparse Random Graphs will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-496267