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Keywords:

  • forbidden subgraphs;
  • hamilton connected

Abstract

Let X, Y be connected graphs. A graph G is inline image-free if G contains a copy of neither X nor Y as an induced subgraph. Pairs of connected graphs inline image such that every 3-connected inline image-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 inline image-free graph is Hamilton connected for inline image and inline image or N1, 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.