Some degree and forbidden subgraph conditions for a graph to have a [formula omitted]-contractible edge

Abstract

An edge in a k-connected graph is said to be k-contractible if the contraction of the edge results in a k-connected graph. We say that a k-connected graph G satisfies “m-degree-sum condition” if ∑x∈V(W)degG(x)≥mk+1 hold for any connected subgraph W of G with |W|=m. Let k be an integer such that k≥5. We prove that if a k-connected graph G with neither K1+C4 nor K2+(K1∪K2) satisfies 3-degree-sum condition, then G has a k-contractible edge. We also prove that if a k-connected graph G with no K1+P4 satisfies 4-degree-sum condition, then G has a k-contractible edge.