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 \(x\).