On Forbidden Pairs Implying Hamilton-Connectedness

• Published in: Journal of graph theory Vol. 72; no. 3; pp. 327 - 345
• Main Authors: Faudree, Jill R., Faudree, Ralph J., Ryjáček, Zdeněk, Vrána, Petr
• Format: Journal Article
• Language: English
• Published: Hoboken Blackwell Publishing Ltd 01.03.2013; Wiley Subscription Services, Inc
• Subjects: forbidden subgraphs; hamilton connected
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.21645

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=K1,3 and Y=P8,N1,1,3, 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.