Vertices of Degree $9$ in Contraction Critical $8$-Connected Graphs

Abstract

In this paper, we have proved that if a contraction critical $8$-connected graph $G$ has no vertices of degree $8$, then for every vertex $x$ of $G$, either $x$ is adjacent to a vertex of degree $9$, or there are at least $4$ vertices of degree $9$ such that every one of them is at distance $2$ from $x$.