On the Rainbow Vertex-Connection
On the Rainbow Vertex-Connection
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DOI：
10.7151/dmgt.1664
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发表时间：
2010-12
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通讯作者：
Xueliang Li;Yongtang Shi
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作者：
Xueliang Li;Yongtang Shi
如果任意两个顶点通过一条内部顶点具有不同颜色的路径相连，则该图为彩虹顶点连通图。连通图G的彩虹顶点连接，用rvc(G)表示，是使G彩虹顶点连接所需的最小颜色数。证明了如果G是最小度为δ的n阶图，则rvc(G) < 11n/δ。本文证明了对于[xxx]和n≥290，rvc(G)≤3n/(δ+1)+5；对于[xxx]， rvc(G)≤4n/(δ +1)+5，对于6≤δ≤15，rvc(G)≤4n/(δ +1)+ C（δ）。我们还证明了当δ = 3时rvc(G)≤3n/4−2，当δ = 4时rvc(G)≤3n/5−8/5，当δ = 5时rvc(G)≤n/2−2。此外，Caro等人构造的一个例子表明，当[xxx]和δ = 3,4,5时，我们的边界被视为紧绷于可加常数。
Abstract 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 vertexconnected. It was proved that if G is a graph of order n with minimum degree δ, then rvc(G) < 11n/δ. In this paper, we show that rvc(G) ≤ 3n/(δ+1)+5 for [xxx] and n ≥ 290, while rvc(G) ≤ 4n/(δ + 1) + 5 for [xxx] and rvc(G) ≤ 4n/(δ + 1) + C(δ) for 6 ≤ δ ≤ 15, where [xxx]. 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 [xxx] and δ = 3, 4, 5, our bounds are seen to be tight up to additive constants.