Sharp concentration of the rainbow connection of random graphs

Yilun Shang
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 16, 2010, Number 4, Pages 25–28
Full paper (PDF, 145 Kb)

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Authors and affiliations

Yilun Shang
Department of Mathematics, Shanghai Jiao Tong University
Shanghai 200240, China

Institute for Cyber Security, University of Texas at San Antonio
San Antonio, Texas 78249, USA

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 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).

Keywords

• Rainbow connection
• Graph coloring
• Concentration
• Random graph

AMS Classification

• 05C80
• 05C15
• 05C40

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