We would like a theorem that says A graph G is hamiltonian if and only if G has property Q, where Q can be checked in polynomial time.

Transcription

1 9 Tough Graphs and Hamilton Cycles We would like a theorem that says A graph G is hamiltonian if and only if G has property Q, where Q can be checked in polynomial time. However in the early 1970 s it was discovered that it is NPcomplete to determine if a graph G has a Hamilton cycle. Hence there is no easy necessary and sufficient condition for a graph to be hamiltonian. Much of the research on Hamilton cycles has focused on finding sufficient conditions for a graph to be hamiltonian. The first and easiest such condition is in the theorem below. Theorem 9.1 (Dirac 1952) Let G be a graph on n 3 verteces. If every vertex is adjacent to at least n/2 vertices, then G is hamiltonian. Note: This condition is best possible, i.e., if (n 1)/2 is substituted for n in Dirac s Thm., it is no longer true. 128

2 Example: Let G = K t,t+1. Then n = 2t + 1 and each vertex has degree at least t = (n 1)/2, but G is not hamiltonian. Note: In G above, if t vertices are removed from G, the graph falls apart into t + 1 components. The graph is not tough enough. Maybe if the graph was more tightly put together it would be forced to be hamiltonian. This motivates the definition of toughness. Let ω(g) denote the number of components of a graph G. S Definition: A graph G is t-tough if t for every ω(g S) subset S of the vertex set V (G) of G with ω(g S) > 1. The toughness of G, denoted τ(g), is the maximum value of t for which G is t-tough (taking τ(k n ) =, for all n 1). To better understand the definition, see the examples. 129

3 Examples: G 1 : t(g 1 ) = 1/4 G 2 : t(g 2 ) = 2/3 G 3 : t(g 3 ) = 1 P : t(p ) = 4/3 130

4 Note: Every hamiltonian graph is 1-tough. However the Petersen graph shows that not every 1-tough graph is hamiltonian. In fact, t(p ) = 4/3. Conjecture:(Chvátal 1973) There exists a finite number t 0 such that τ(g) t 0 G is hamiltonian. Question: Is this true, and if so, what is the smallest value of t 0? 131

5 Definition: A 2-factor of a graph G is a subgraph in which every vertex has degree 2. Note: This means that every hamiltonian graph has a 2-factor, but not every graph with a 2-factor is hamiltonian. Example: A disconnected graph consisting of two disjoint triangles has a 2-factor, but is clearly not hamiltonian. There are also examples that are connected. Example: The Petersen graph P has a 2-factor, but is not hamiltonian. Theorem 9.2 (Chvátal 1973) For every ɛ > 0 there exist (3/2 ɛ) - tough graphs with no 2-factor. Theorem 9.3 (Chvátal 1973) There exist infinitely many 3/2 - tough nonhamiltonian graphs. Maybe all graphs with τ(g) > 3/2 are hamiltonian. 132

6 Theorem 9.4 (Thomassen) There exist infinitely many nonhamiltonian graphs G with τ(g) > 3/2. The next result led to the 2-tough conjecture. Theorem 9.5 (Enomoto, Jackson, Katerinis, Saito 1985) All 2-tough graphs have 2-factors. Furthermore, for any ɛ > 0, there exists a (2 ɛ) - tough graph with no 2-factor. Note: This means that the smallest possible value of t 0 such that all t 0 - tough graphs are hamiltonian is t 0 = 2. In addition, there were many other reasons why the following became a very intriguing conjecture. Conjecture: τ(g) 2 G is hamiltonian. Suppose this conjecture is true. It was hoped that at least the hypothesis could be checked in polynomial time. This is the case in theorems like Dirac s Theorem. 133

7 Unfortunately, it s not true for toughness. Theorem 9.6 (Bauer, Hakimi, Schmeichel 1990) For any rational t > 0, it is NP-hard to determine if a graph is t-tough. Nevertheless, there was still a lot of interest in the 2-tough conjecture. Theorem 9.7 (Bauer, Broersma, Veldman 2000) For any ɛ > 0 there exists a (9/4 ɛ) - tough nonhamiltonian graph. See figure. Note: It is still not known if there exists a finite constant t 0 such that every t 0 -tough graph is hamiltonian. 134

8 n = 42 G: t(g) = 2 - NOT HAMILTONIAN 135

9 G - NOT HAMILTONIAN G : t(g ) = 4(2t + 1) + t 2(2t + 1) + 1 = 9t + 4 4t

10 Definition: A graph G is chordal if it contains no induced cycle of length at least four. Theorem 9.8 (Bauer, Katona, Kratsch, Veldman 2000) Let G be a 3 2-tough chordal graph. Then G has a 2-factor. Note: Graphs in Chvátal s 1973 paper show that this result is best possible. Recently Chen, Jacobson, Kézdy and Lehel (1997) have shown that every 18-tough chordal graph on n 3 vertices is hamiltonian. We believe a stronger result is true; namely that every 2-tough chordal graph is hamiltonian. We also believe that for every ɛ > 0 there exists a (2 ɛ)-tough chordal nonhamiltonian graph. We have already taken a step in this direction. Theorem 9.9 For every ɛ > 0 there exists a ( 7 4 ɛ)-tough chordal nontraceable graph. 137

11 H - CHORDAL AND NOT HAMILTONIAN H: t(h) = 3(2t + 1) + t 2(2t + 1) + 1 = 7t + 3 4t

12 Conjecture: If G is a maximum planar graph with τ(g) = 3, then G has a 2-factor. 2 Why is this interesting? Let ɛ > 0. Note 1: If τ(g) = ɛ, then the conjecture is true. This follows from the fact that a ( connected. + ɛ)-tough graph must be Theorem 9.10 (Tutte 1956) Every 4-connected planar graph is hamiltonian. Note 2: If τ(g) = 3 2 ɛ, then the conjecture is false. In 1999, P. Owens found a collection of maximum planar ( 3 2 ɛ)-tough nonhamiltonian graphs. Upon inspection, it was clear that these graphs have no 2-factor. 139

13 Two questions 1. Are all ( 3 )-tough maximum planar graphs hamiltonian? 2 2. Do all ( 3 )-tough planar graphs have a 2-factor? 2 Note: If our conjecture is false, the answer to each question is No. If our conjecture is true,???????????????? Note: There exist ( 3 )-tough planar nonhamiltonian graphs