Hypohamiltonian graph

In the mathematical field of graph theory, a graph "G" is said to be hypohamiltonian if "G" does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from "G" is Hamiltonian.

History

harvtxt|Grötschel|1980 sums up much of the research in this area with the following sentence: “The articles dealing with those graphs ... usually exhibit new classes of hypohamiltonian or hypotraceable graphs showing that for certain orders "n" such graphs indeed exist or that they possess strange and unexpected properties.”

Applications

Hypohamiltonian graphs arise in integer programming solutions to the traveling salesman problem: certain kinds of hypohamiltonian graphs define facets of the "traveling salesman polytope", a shape defined as the convex hull of the set of possible solutions to the traveling salesman problem, and these facets may be used in cutting-plane methods for solving the problem. [harvtxt|Grötschel|1977; harvtxt|Grötschel|1980; harvtxt|Grötschel|Wakabayashi|1981.] harvtxt|Grötschel|1980 observes that the computational complexity of determining whether a graph is hypohamiltonian, although unknown, is likely to be high, making it difficult to find facets of these types except for those defined by small hypohamiltonian graphs; fortunately, the smallest graphs lead to the strongest inequalities for this application. [harvtxt|Goemans|1995.]

Concepts closely related to hypohamiltonicity have also been used by harvtxt|Park|Lim|Kim|2007 to measure the fault tolerance of network topologies for parallel computing.

Properties

Every hypohamiltonian graph must be 3-vertex-connected, as the removal of any two vertices leaves a Hamiltonian path, which is connected. There exist "n"-vertex hypohamiltonian graphs in which the maximum degree is "n"/2, and in which there are approximately "n"²/4 edges. [harvtxt|Thomassen|1981.]

harvtxt|Herz|Duby|Vigué|1967 conjectured that every hypohamiltonian graph has girth 5 or more, but this was disproved by harvtxt|Thomassen|1974b, who found examples with girth 3 and 4. For some time it was unknown whether a hypohamiltonian graph could be planar, but several examples are now known, [The existence of planar hypohamiltonian graphs was posed as an open question by harvtxt|Chvátal|1973, and harvtxt|Chvátal|Klarner|Knuth|1972 offered a $5 prize for the construction of one. harvtxt|Thomassen|1976 found planar hypohamiltonian graphs of girth 3, 4, and 5 and showed that there exist infinitely many planar hypohamiltonian graphs.] the smallest of which has 48 vertices. [harvtxt|Zamfirescu|Zamfirescu|2007. An earlier small planar hypohamiltonian graph with 57 vertices was found by harvtxt|Hatzel|1979.] Every planar hypohamiltonian graph has at least one vertex with only three incident edges. [harvtxt|Thomassen|1978.]

If a 3-regular graph is Hamiltonian, its edges can be colored with three colors: use alternating colors for the edges on the Hamiltonian cycle (which must have even length by the handshaking lemma) and a third color for all remaining edges. Therefore, all snarks, bridgeless cubic graphs that require four edge colors, must be non-Hamiltonian, and many known snarks are hypohamiltonian. Every hypohamiltonian snark is "bicritical": removing any two vertices leaves a subgraph the edges of which can be colored with only three colors. [harvtxt|Steffen|1998; harvtxt|Steffen|2001.] A three-coloring of this subgraph can be simply described: after removing one vertex, the remaining vertices contain a Hamiltonian cycle. After removing a second vertex, this cycle becomes a path, the edges of which may be colored by alternating between two colors. The remaining edges form a matching and may be colored with a third color.

The color classes of any 3-coloring of the edges of a 3-regular graph form three matchings such that each edge belongs to exactly one of the matchings.Hypohamiltonian snarks do not have a partition into matchings of this type, but harvtxt|Häggkvist|2007 conjectures that the edges of any hypohamiltonian snark may be used to form six matchings such that each edge belongs to exactly two of the matchings. This is a special case of the Berge–Fulkerson conjecture that any snark has six matchings with this property.

Hypohamiltonian graphs cannot be bipartite: in a bipartite graph, a vertex can only be deleted to form a Hamiltonian subgraph if it belongs to the larger of the graph's two color classes. However, every bipartite graph occurs as an induced subgraph of some hypohamiltonian graph. [harvtxt|Collier|Schmeichel|1977.]

Examples

The smallest hypohamiltonian graph is the Petersen graph harv|Herz|Duby|Vigué|1967. More generally, the generalized Petersen graph GP("n",2) is hypohamiltonian when "n" is 5 (mod 6); [harvtxt|Robertson|1969 proved that these graphs are non-Hamiltonian, while it is straightforward to verify that their one-vertex deletions are Hamiltonian. See harvtxt|Alspach|1983 for a classificiation of non-Hamiltonian generalized Petersen graphs.] the Petersen graph is the instance of this construction with "n" = 5.

harvtxt|Lindgren|1967 found another infinite class of cubic graphs in which the number of vertices is 4 (mod 6). Lindgren's construction consists of a cycle of length 3 (mod 6) and a single central vertex; the central vertex is connected to every third vertex of the cycle by edges he calls spokes, and the vertices two positions away from each spoke endpoint are connected to each other. Again, the smallest instance of Lindgren's construction is the Petersen graph.

Enumeration

Václav Chvátal proved in 1973 that for all sufficiently large "n" there exists a hypohamiltonian graph with "n" vertices. Taking into account subsequent discoveries, [harvtxt|Thomassen|1974a; harvtxt|Doyen|van Diest|1975.] “sufficiently large” is now known to mean that such graphs exist for all "n" ≥ 18. A complete list of hypohamiltonian graphs with at most 17 vertices is known: [harvtxt|Aldred|McKay|Wormald|1997. See also OEIS|id=A141150.] they are the 10-vertex Petersen graph, a 13-vertex graph and a 15-vertex graph found by computer searches of harvtxt|Herz|1968, and four 16-vertex graphs. There exist at least thirteen 18-vertex hypohamiltonian graphs. By applying the flip-flop method of harvtxt|Chvátal|1973 to the Petersen graph and the flower snark, it is possible to show that the number of hypohamiltonian graphs, and more specifically the number of hypohamiltonian snarks, grows as an exponential function of the number of vertices. [harvtxt|Skupień|1989; harvtxt|Skupień|2007.]

Generalizations

An equivalent definition of hypohamiltonian graphs is that their longest cycle has length "n" − 1 and that the intersection of all longest cycles is empty. harvtxt|Menke|Zamfirescu|Zamfirescu|1998 investigate graphs with the same property that the intersection of longest cycles is empty but in which the longest cycle length is shorter than "n" − 1. harvtxt|Herz|1968 defines the "cyclability" of a graph as the largest number "k" such that every "k" vertices belong to a cycle; the hypohamiltonian graphs are exactly the graphs that have cyclability "n" − 1. Similarly, harvtxt|Park|Lim|Kim|2007 define a graph to be &fnof;-"fault hamiltonian" if the removal of at most &fnof; vertices leaves a Hamiltonian subgraph. harvtxt|Schauerte|Zamfirescu|2006 study the graphs with cyclability "n" − 2.

Notes

References

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hypohamiltonian — adjective Of a graph, not containing a Hamiltonian cycle but such that the removal of any single vertex produces a Hamiltonian graph … Wiktionary
Petersen graph — Infobox graph name = Petersen graph image caption = The Petersen graph is most commonly drawn as a pentagon with a pentagram inside, with five spokes. namesake = Julius Petersen vertices = 10 edges = 15 radius = 2 diameter = 2 girth = 5 chromatic … Wikipedia
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Snark (graph theory) — In graph theory, a snark is a connected, bridgeless cubic graph with chromatic index equal to 4. In other words, it is a graph in which every vertex has three neighbors, and the edges cannot be colored by three colors without two edges of the… … Wikipedia
Graphe hypohamiltonien — En théorie des graphes, un graphe est hypohamiltonien s il n a pas de cycle hamiltonien mais que la suppression de n importe quel sommet du graphe suffit à le rendre hamiltonien. Sommaire 1 Histoire 2 Planarité 3 Exemples … Wikipédia en Français
Double-star snark — The Double star snark Vertices 30 Edges 45 Chromatic number … Wikipedia
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