J.A. Bondy U.S.R. Murty Graph Theory (III) ABC J.A. Bondy, PhD U.S.R. Murty, PhD Universite´ Claude-Bernard Lyon 1 Mathematics Faculty Domaine de Gerland University of Waterloo 50 Avenue Tony Garnier 200 University Avenue West 69366 Lyon Cedex 07 Waterloo, Ontario, Canada France N2L 3G1 Editorial Board S. Axler K.A. Ribet Mathematics Department Mathematics Department San Francisco State University University of California, Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA Graduate Texts in Mathematics series ISSN: 0072-5285 ISBN: 978-1-84628-969-9 e-ISBN: 978-1-84628-970-5 DOI: 10.1007/978-1-84628-970-5 Library of Congress Control Number: 2007940370 Mathematics Subject Classification (2000): 05C; 68R10 °c J.A. Bondy & U.S.R. 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Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com Contents 1 Graphs ........................................................ 1 2 Subgraphs ..................................................... 39 3 Connected Graphs ............................................. 79 4 Trees .......................................................... 99 5 Nonseparable Graphs ..........................................117 6 Tree-Search Algorithms ........................................135 7 Flows in Networks .............................................157 8 Complexity of Algorithms .....................................173 9 Connectivity ...................................................205 10 Planar Graphs .................................................243 11 The Four-Colour Problem .....................................287 12 Stable Sets and Cliques ........................................295 13 The Probabilistic Method .....................................329 14 Vertex Colourings .............................................357 15 Colourings of Maps ............................................391 16 Matchings .....................................................413 17 Edge Colourings ...............................................451 XII Contents 18 Hamilton Cycles ...............................................471 19 Coverings and Packings in Directed Graphs ...................503 20 Electrical Networks ............................................527 21 Integer Flows and Coverings ...................................557 Unsolved Problems ................................................583 References .........................................................593 General Mathematical Notation ...................................623 Graph Parameters .................................................625 Operations and Relations ..........................................627 Families of Graphs .................................................629 Structures .........................................................631 Other Notation ....................................................633 Index ..............................................................637 18 Hamilton Cycles Contents 18.1 Hamiltonian and Nonhamiltonian Graphs ..........471 Tough Graphs ....................................... 472 Hypohamiltonian Graphs ............................ 473 18.2 Nonhamiltonian Planar Graphs . ...................478 Grinberg’s Theorem ................................. 478 Barnette’s Conjecture .............................. 481 18.3 Path and Cycle Exchanges ........................483 Path Exchanges ..................................... 484 Cycle Exchanges .................................... 485 Dirac’s Theorem ..................................... 485 The Closure of a Graph ............................. 486 The Chvatal–Erd´ os˝ Theorem ........................ 488 18.4 Path Exchanges and Parity . .......................492 The Lollipop Lemma ................................. 492 Uniquely Hamiltonian Graphs ....................... 494 Sheehan’s Conjecture ............................... 494 18.5 Hamilton Cycles in Random Graphs ...............499 Posa’s´ Lemma ........................................ 499 18.6 Related Reading . .................................501 The Bridge Lemma ................................... 501 The Hopping Lemma ................................. 502 Long Paths and Cycles .............................. 502 18.1 Hamiltonian and Nonhamiltonian Graphs Recall that a path or cycle which contains every vertex of a graph is called a Hamilton path or Hamilton cycle of the graph. Such paths and cycles are named after Sir William Rowan Hamilton, who described, in a letter to his friend Graves in 1856, a mathematical game on the dodecahedron (Figure 18.1a) in which one 472 18 Hamilton Cycles person sticks pins in any five consecutive vertices and the other is required to com- plete the path so formed to a spanning cycle (see Biggs et al. (1986) or Hamilton (1931)). Hamilton was prompted to consider such cycles in his early investigations into group theory, the three edges incident to a vertex corresponding to three generators of a group. Agraphistraceable if it contains a Hamilton path, and hamiltonian if it con- tains a Hamilton cycle. The dodecahedron is hamiltonian; a Hamilton cycle is indicated in Figure 18.1a. On the other hand, the Herschel graph of Figure 18.1b is nonhamiltonian, because it is bipartite and has an odd number of vertices. This graph is, however, traceable. (a) (b) Fig. 18.1. Hamiltonian and nonhamiltonian graphs: (a) the dodecahedron, (b) the Her- schel graph Tough Graphs As we saw in Section 8.3, the problem of deciding whether a given graph is hamil- tonian is NP-complete. It is therefore natural to look for reasonable necessary or sufficient conditions for the existence of Hamilton cycles. The following simple necessary condition turns out to be surprisingly useful. Theorem 18.1 Let S be a set of vertices of a hamiltonian graph G.Then c(G − S) ≤|S| (18.1) Moreover, if equality holds in (18.1), then each of the |S| components of G − S is traceable, and every Hamilton cycle of G includes a Hamilton path in each of these components. Proof Let C be a Hamilton cycle of G. Then C − S clearly has at most |S| components. But this implies that G−S also has at most |S| components, because C is a spanning subgraph of G. 18.1 Hamiltonian and Nonhamiltonian Graphs 473 If G − S has exactly |S| components, C − S also has exactly |S| components, and the components of C − S are spanning subgraphs of the components of G − S. In other words, C includes a Hamilton path in each component of G − S. AgraphG is called tough if (18.1) holds for every nonempty proper subset S of V . By Theorem 18.1, a graph which is not tough cannot be hamiltonian. As an illustration, consider the graph G of Figure 18.2a. This graph has nine vertices. On deleting the set S of three vertices indicated, four components remain. This shows that the graph is not tough, and we infer from Theorem 18.1 that it is nonhamiltonian. Although condition (18.1) has a simple form, it is not always easy to apply. In fact, as was shown by Bauer et al. (1990), recognizing tough graphs is NP-hard. (a) (b) Fig. 18.2. (a) A nontough graph G, (b) the components of G − S Hypohamiltonian Graphs As the above example shows, Theorem 18.1 can sometimes be applied to deduce that a graph is nonhamiltonian. Such an approach does not always work. The Petersen graph is nonhamiltonian (Exercises 2.2.6, 17.1.8), but one cannot deduce this fact from Theorem 18.1. Indeed, the Petersen graph has a very special prop- erty: not only is it nonhamiltonian, but the deletion of any one vertex results in a hamiltonian graph (Exercise 18.1.16a). Such graphs are called hypohamiltonian. Deleting a single vertex from a hypohamiltonian graph results in a subgraph with just one component, and deleting a set S of at least two vertices produces no more than |S|−1 components, because each vertex-deleted subgraph is hamiltonian, hence tough. The Petersen graph is an example of a vertex-transitive hypohamil- tonian graph. Such graphs appear to be extremely rare. Another example is the Coxeter graph (see Exercises 18.1.14 and 18.1.16c); the attractive drawing of this graph shown in Figure 18.3 is due to Rand´ıc(1981). Its geometric origins and many of its interesting properties are described in Coxeter (1983). 474 18 Hamilton Cycles Fig. 18.3. The Coxeter graph Exercises 18.1.1 By applying Theorem 18.1, show that the Herschel graph (Figure 18.1b) is nonhamiltonian. (It is, in fact, the smallest nonhamiltonian 3-connected planar graph.) 18.1.2 Let G be a cubic graph, and let H be the cubic graph obtained from G by expanding a vertex to a triangle. Exhibit a bijection between the Hamilton cycles