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- W2806198738 abstract "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 $om=om(n)toinfty$ 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 $rc(G) sim maxset{Z_1,diameter(G)}$ 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." @default.
- W2806198738 created "2018-06-13" @default.
- W2806198738 creator A5056950019 @default.
- W2806198738 creator A5069421706 @default.
- W2806198738 date "2012-01-22" @default.
- W2806198738 modified "2023-09-27" @default.
- W2806198738 title "Rainbow Connectivity of $G(n,p)$ at the connectivity threshold" @default.
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