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 n2/4 edges.
Herz, Duby & Vigué (1967) conjectured that every hypohamiltonian graph has girth 5 or more, but this was disproved by 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 smallest of which has 40 vertices. Every planar hypohamiltonian graph has at least one vertex with only three incident edges.
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. 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 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.
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