Show that if u and v are nonadjacent vertices in a graph G...

Show that if $u$ and $v$ are nonadjacent vertices in a graph $G$ with $n$ vertices and $\operatorname{deg}(u)+\operatorname{deg}(v) \geq n,$ then $G$ has a Hamilton circuit if and only if $G+\{u, v\}$ has a Hamilton circuit.

Key Concepts

Hamilton Circuit

A Hamilton circuit is a cycle in a graph that visits each vertex exactly once before returning to the starting vertex. It is a fundamental concept in graph theory, used to study the traversal properties of graphs and is central to problems involving complete coverage of a graph's vertices.

Graph Augmentation

Graph augmentation involves adding edges (or vertices) to a graph with the goal of altering or improving its properties. In contexts related to Hamiltonicity, specifically adding an edge between two nonadjacent vertices can be instrumental in forming or verifying the existence of a Hamilton circuit by potentially bridging gaps in the cycle.

Degree Condition

The degree condition refers to criteria based on the number of edges incident to vertices in a graph. When studying Hamilton circuits, a common requirement is that the sum of the degrees of a pair of nonadjacent vertices meets or exceeds a certain threshold (such as the total number of vertices). This condition provides sufficient structure to argue for the existence of a Hamilton circuit.

Ore's Theorem

Ore's Theorem asserts that a graph with n vertices (n ? 3) is Hamiltonian if, for every pair of nonadjacent vertices, the sum of their degrees is at least n. This theorem generalizes Dirac's theorem and plays a crucial role in establishing Hamiltonicity in graphs under given degree conditions, linking local vertex properties to global cycle structure.

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