Contractible Edges in k-Connected Graphs with Some Forbidden Subgraphs

Abstract

In 2001, Kawarabayashi proved that for any odd integer k ? 3, if a k-connected graph G is $${K^{-}_{4}}$$ K 4 - -free, then G has a k-contractible edge. He pointed out, by a counterexample, that this result does not hold when k is even. In this paper, we have proved the following two results on the subject: (1) For any even integer k ? 4, if a k-connected graph G is $${K_{4}^{-}}$$ K 4 - -free and d G (x) + d G (y) ? 2k + 1 hold for every two adjacent vertices x and y of V(G), then G has a k-contractible edge. (2) Let t ? 3, k ? 2t --- 1 be integers. If a k-connected graph G is $${(K_{1}+(K_{2} \cup K_{1, t}))}$$ ( K 1 + ( K 2 ? K 1 , t ) ) -free and d G (x) + d G (y) ? 2k + 1 hold for every two adjacent vertices x and y of V(G), then G has a k-contractible edge.