数学季刊 ›› 2016, Vol. 31 ›› Issue (4): 399-405.doi: 10.13371/j.cnki.chin.q.j.m.2016.04.008
摘要: A vertex-colored graph G is said to be rainbow vertex-connected if every two vertices of G are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number 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. If for every pair u, v of distinct vertices, G contains a rainbow u-v geodesic, then G is strong rainbow vertex-connected. The minimum number k for which there exists a k-vertex-coloring of G that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of G, denoted by srvc(G). Observe that rvc(G) ≤ srvc(G) for any nontrivial connected graph G. In this paper, for a Ladder Ln,we determine the exact value of srvc(Ln) for n even. For n odd, upper and lower bounds of srvc(Ln) are obtained. We also give upper and lower bounds of the(strong) rainbow vertex-connection number of Mbius Ladder.
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