Rainbow Vertex-connection Number of Ladder and MÄobius Ladder

LIU Hui-min, MAO Ya-ping

doi:10.13371/j.cnki.chin.q.j.m.2016.04.008

Chinese Quarterly Journal of Mathematics >

2016 , Vol. 31 >Issue 4: 399 - 405

DOI: https://doi.org/10.13371/j.cnki.chin.q.j.m.2016.04.008

Rainbow Vertex-connection Number of Ladder and MÄobius Ladder

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Department of Mathematics,Qinghai Normal University.

LIU Hui-min(1969-), female, native of Nanyang, Henan, an associate professor of Qinghai Normal University, M.S.D., engages in graph theory and combinatorial optimization.

Received date: 2014-07-31

Online published: 2020-11-03

Supported by

Supported by the National Natural Science Foundation of China(11551001,11061027,11261047,11161037,11461054)； Supported by the Science Found of Qinghai Province(2016-ZJ-948Q,2014-ZJ-907)；

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Abstract

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（L_n） for n even. For n odd, upper and lower bounds of srvc（L_n） are obtained. We also give upper and lower bounds of the（strong） rainbow vertex-connection number of Mbius Ladder.

Key words： vertex-coloring; rainbow vertex-connection; (strong) rainbow vertex-connection number; Ladder; M?obius Ladder

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LIU Hui-min, MAO Ya-ping . Rainbow Vertex-connection Number of Ladder and MÄobius Ladder[J]. Chinese Quarterly Journal of Mathematics, 2016 , 31(4) : 399 -405 . DOI: 10.13371/j.cnki.chin.q.j.m.2016.04.008

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