More than one tough chordal planar graphs are hamiltonian.pdfVIP

More than One ToughChordal PlanarGraphs are HamiltonianThomas BohmeJochen HarantDEPARTMENT OF MATHEMATICSTECHNICAL UNIVERSITY OF ILMENAUD-98684 ILMENAU, PF 100 565, GERMANYMichal TkacDEPARTMENT OF MATHEMATICSTECHNICAL UNIVERSITY OF KOSICE042 00 KOSICE, SLOVAKIAABSTRACTWe prove the result stated in the title. Furthermore, it is proved that for any 0 thereis a 1-tough chordal planar graph G such that the length of a longest cycles of G isless than jV (G)j. c 1994 John Wiley Sons, Inc.Keywords: Hamiltonian cycles, chordal planar graphs, toughness1. INTRODUCTION AND NOTATIONLet G be a connected graph. A subset S of V (G) separates G if the graph G S ob-tained from G by deleting the vertices of S is disconnected. G is said to be k-tough iffor every separating set S  V (G) the number !(G S) of components of G S is atmost kjSj. The toughness (G) of a non-complete graph G is de ned to be the largestinteger t 0 such that G is t-tough. For a complete graph G we let (G) = 1. Theconcept of toughness was introduced by Chvatal [4] in 1973. It is easy to see that everyHamiltonian graph is 1-tough. In [4] Chvatal made the following conjecture.D R A F T March 19, 1998, 1:17pm D R A F T 482 JOURNAL OF GRAPH THEORYConjecture (Chvatal [4]) There is a t0 0 such that every t0-tough graph is Ha-miltonian.In [4] also the stronger conjecture was made that every 32 -tough graph is Hamiltonian.This conjecture was rst disproved by Thomassen (see [2]). Enomoto, Jackson, Katerinisand Saito [7] showed that every 2-tough graph has a 2-factor (a 2-regular spanningsubgraph) while for every 0 there is a (2 )-tough graph without a 2-factor.For a long time it was conjectured that every 2-tough graph is Hamiltonian. Thisconjecture remained open until recently, Bauer, Broersma and Veldman [1] disproved itby constructing a ( 94 )-tough non-Hamiltonian graph for arbitrary 0.More is known about the existence of Hamiltonian cycles in planar and chordal graphs.It follows from a