On the Rainbow Vertex-Connection

Li, Xueliang ; Shi, Yongtang

Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 307-313

Résumé

A vertex-colored graph 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. It was proved that if G is a graph of order n with minimum degree δ, then rvc(G) lt; 11n//δ. In this paper, we show that rvc(G) ≤ 3n//(δ+1)+5 for δ≥√(n-1) -1 and n ≤ 290, while rvc(G) ≤ 4n//(δ + 1) + 5 for 16 ≤δ≤√(n-1)-2 and rvc(G) ≤ 4n//(δ + 1) + C(δ) for 6 ≤δ≤ 15, where C(δ) = e^ 3 log (δ^3 + 2 δ^2 +3)-3(log 3 - 1)/δ - 3 - 2. We also prove that rvc(G) ≤ 3n//4 − 2 for δ = 3, rvc(G) ≤ 3n//5 − 8//5 for δ = 4 and rvc(G) ≤ n//2 − 2 for δ = 5. Moreover, an example constructed by Caro et al. shows that when δ≥√(n-1) - 1 and δ = 3, 4, 5, our bounds are seen to be tight up to additive constants.

Détail

Keywords: rainbow vertex-connection, vertex coloring, minimum degree, 2-step dominating set