Some properties of contraction-critical 5-connected graphs Original Research Article

Let image be a positive integer and let image be a image-connected graph. An edge of image is called image-contractible if its contraction still results in a image-connected graph. A non-complete image-connected graph image is called contraction-critical if image has no image-contractible edge. Let image 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 image is adjacent to at least two vertices of degree 5, and thus image has at least image 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 image. As a result, we prove that a contraction-critical 5-connected graph image has at least image vertices of degree 5.