## Sharp Concentration of the Rainbow Connection of Random Graphs

**Authors:** Yilun Shang

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. Similarly, a vertex-colored graph G is rainbow
vertex-connected if any two vertices are connected by a path whose internal vertices
have distinct colors. The rainbow vertex-connection of a connected graph G, denoted
by rvc(G), is the smallest number of colors that are needed in order to make G rainbow
vertex-connected. We prove that both rc(G) and rvc(G) have sharp concentration in
classical random graph model G(n, p).

**Comments:** 5 pages

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### Submission history

[v1] 28 Jan 2011

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