Disconnected Forbidden Subgraphs, Toughness and Hamilton Cycles

Zh. G. Nikoghosyan

doi:10.1155/2013/673971

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.

Highlights

Finpite undiprected graphs without loops or multiple edges are considered
The graph K1,3 + e is obtained from K1,3 by adding an edge
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

Summary

Introduction

Finpite undiprected graphs without loops or multiple edges are considered. Let G be a graph with vertex set V(G) and edge set E(G). The proposition shows that (K1∪K1∪K1)-free graphs are hamiltonian if and only if they are 2-connected. To describe the hamiltonian graphs with forbidden subgraph (K1 ∪ P2), we need the following recursive definition. The above constructed graph H1 shows that the condition “G is (K1 ∪ P4)-free” in Conjecture 7 cannot be replaced by “G is (K1 ∪ P5)-free.”. The graph H1 (see the graph examples concerning the sharpness of Theorem 6) shows that the condition τ > 1 in Conjecture 10 can not be replaced by τ = 1. The Petersen graph shows that the condition “G is (K2 ∪ K2)-free” in Conjecture 10 can not be replaced by “G is (K2 ∪ K3)-free”.

Additional Notation and Preliminaries

Proofs