The k-subconnectedness of planar graphs

Abstract

<abstract> A graph $ G $ with at least $ 2k $ vertices is called k-subconnected if, for any $ 2k $ vertices $ x_{1}, x_{2}, \cdots, x_{2k} $ in $ G $, there are $ k $ independent paths joining the $ 2k $ vertices in pairs in $ G $. In this paper, we prove that a k-connected planar graph with at least $ 2k $ vertices is k-subconnected for $ k = 1, 2 $; a 4-connected planar graph is k-subconnected for each $ k $ such that $ 1\leq k\leq \nu /2 $, where $ v $ is the number of vertices of $ G $; and a 3-connected planar graph $ G $ with at least $ 2k $ vertices is k-subconnected for $ k = 4, 5, 6 $. The bounds of k-subconnectedness are sharp. </abstract>