Some properties of contraction-critical 5-connected graphs

Abstract

Let k be a positive integer and let G be a k -connected graph. An edge of G is called k -contractible if its contraction still results in a k -connected graph. A non-complete k -connected graph G is called contraction-critical if G has no k -contractible edge. Let G be a contraction-critical 5-connected graph, Su proved in [J. Su, Vertices of degree 5 in contraction-critical 5-connected graphs, J. Guangxi Normal Univ. 17 (3) (1997) 12–16 (in Chinese)] that each vertex of G is adjacent to at least two vertices of degree 5, and thus G has at least 2 5 | V ( G ) | vertices of degree 5. In this paper, we further study the properties of contraction-critical 5-connected graph. In the process, we investigate the structure of the subgraph induced by the vertices of degree 5 of G . As a result, we prove that a contraction-critical 5-connected graph G has at least 4 9 | V ( G ) | vertices of degree 5.