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A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path.
Hamiltonian if it has a Hamiltonian cycle. Closure: The (Hamiltonian) closure of a graph G, denoted Cl(G), is the simple graph obtained from G by repeatedly adding edges joining pairs of nonadjacent vertices with degree sum at least jV(G)j until no such pair remains. Lemma 10.1 A graph G is Hamiltonian if and only if its closure is Hamiltonian.
The point of defining the closure is that it enables us to state the following lovely result: Theorem 11.7 (Bondy and Chvátal, 1976). A graph G is Hamiltonian if and only if its closure [G] is Hamiltonian. Before we prove this, notice that Dirac’s and Ore’s theorems are easy corollaries,
Prove a simple graph $G$ is Hamiltonian if and only if its closure is Hamiltonian. $|V(G)|=n$, $\deg(u)+\deg(v)\ge n$ ($u$ and $v$ is a pair of non-adjacent vertices on $G$) I know if $G$ is Hamiltonian, its closure must be Hamiltonian.
21 de nov. de 2014 · A school course in graph theory.
Definition: Closure. Let \(G\) be a graph on \(n\) vertices. The closure of \(G\) is the graph obtained by repeatedly joining pairs of nonadjacent vertices \(u\) and \(v\) for which \(d(u) + d(v) ≥ n\), until no such pair exists.
A Hamiltonian path is a traversal of a (finite) graph that touches each vertex exactly once. If the start and end of the path are neighbors (i.e. share a common edge), the path can be extended to a cycle called a Hamiltonian cycle .
