Almost All 3-Connected Graphs Contain a Contractible Set of k Vertices

McCuaig and Ota conjectured that every sufficiently large 3-connected graph G contains a connected subgraph H on k vertices such that G?V(H) is 2-connected. We prove the weaker statement that every sufficiently large 3-connected graph G contains a not necessarily connected subgraph H on k vertices such that G?V(H) is 2-connected.

[1]  Elwood S. Buffa,et al.  Graph Theory with Applications , 1977 .

[2]  Matthias Kriesell,et al.  Contractible Subgraphs in 3-Connected Graphs , 2000, J. Comb. Theory, Ser. B.

[3]  Christopher A. Rodger,et al.  (k, G)-cages Are 3-connected , 1999, Discret. Math..

[4]  Katsuhiro Ota,et al.  Contractile Triples in 3-Connected Graphs , 1994, J. Comb. Theory, Ser. B.

[5]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[6]  William T. Tutte,et al.  A theory of 3-connected graphs , 1961 .

[7]  Frank Harary,et al.  Graph Theory , 2016 .