Table of Contents
ISRN Combinatorics
Volume 2013, Article ID 673971, 4 pages
http://dx.doi.org/10.1155/2013/673971
Research Article

Disconnected Forbidden Subgraphs, Toughness and Hamilton Cycles

Institute for Informatics and Automation Problems, National Academy of Sciences of the Republic of Armenia, 1 P. Sevak Street, 0014 Yerevan, Armenia

Received 2 October 2012; Accepted 18 November 2012

Academic Editors: M. Alekseyev, M. Eliasi, N. A. Gordon, A. V. Kelarev, C.-K. Lin, V. Y. Protasov, and Y. Zhang

Copyright © 2013 Zh. G. Nikoghosyan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In 1974, Goodman and Hedetniemi proved that every 2-connected -free graph is hamiltonian. This result gave rise many other conditions for Hamilton cycles concerning various pairs and triples of forbidden connected subgraphs under additional connectivity conditions. In this paper we investigate analogous problems when forbidden subgraphs are disconnected which affects more global structures in graphs such as tough structures instead of traditional connectivity structures. In 1997, it was proved that a single forbidden connected subgraph in 2-connected graphs can create only a trivial class of hamiltonian graphs (complete graphs) with . In this paper we prove that a single forbidden subgraph can create a non trivial class of hamiltonian graphs if is disconnected: every -free graph either is hamiltonian or belongs to a well defined class of non hamiltonian graphs; every 1-tough -free graph is hamiltonian. We conjecture that every 1-tough -free graph is hamiltonian and every 1-tough -free graph is hamiltonian.