On Forbidden Pairs Implying Hamilton‐Connectedness

Let X , Y be connected graphs. A graph G is ( X , Y ) ‐free if G contains a copy of neither X nor Y as an induced subgraph. Pairs of connected graphs X , Y such that every 3‐connected ( X , Y ) ‐free graph is Hamilton connected have been investigated most recently in (Guantao Chen and Ronald J. Gould, Bull. Inst. Combin. Appl., 29 (2000), 25–32.) [8] and (H. Broersma, R. J. Faudree, A. Huck, H. Trommel, and H. J. Veldman, J. Graph Theory, 40(2) (2002), 104–119.) [5]. This paper improves those results. Specifically, it is shown that every 3‐connected ( X , Y ) ‐free graph is Hamilton connected for X = K 1 , 3and Y = P 8 ,N 1 , 1 , 3 , or N 1, 2, 2 and the proof of this result uses a new closure technique developed by the third and fourth authors. A discussion of restrictions on the nature of the graph Y is also included.