-=Let G be a graph of order n having at least ((n-1)(n-2))/(2)+2 edges, Prove that G has a Hamil

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-=Let G be a graph of order n having at least...

-=Let $G$ be a graph of order $n$ having at least
$$
\frac{(n-1)(n-2)}{2}+2
$$
edges, Prove that $G$ has a Hamilton cycle. Exhibit a graph of order $n$ with one fewer edge that does not have a Hamilton cycle.

Key Concepts

Hamiltonian Cycle

A Hamiltonian cycle in a graph is a cycle that visits every vertex exactly once and returns to the starting vertex. This concept is central to many problems in graph theory and combinatorics and has implications in routing, scheduling, and the study of network connectivity.

Edge Count Conditions

Edge count conditions refer to establishing thresholds on the number of edges a graph must have in relation to its number of vertices to guarantee certain properties such as the existence of a Hamiltonian cycle. These conditions often stem from classical theorems like Dirac’s and Ore’s, which relate vertex degrees or total edge count to Hamiltonicity.

Extremal Graph Theory

Extremal graph theory deals with problems that find the extreme (minimal or maximal) number of edges or other graph parameters required for a graph to exhibit a particular property. In the context of Hamiltonian cycles, it involves determining the minimum number of edges that guarantee the existence of a Hamiltonian cycle in a graph of a given order.

Sharpness of Results

The sharpness of a result refers to whether the conditions provided (such as an edge count threshold) are the best possible, meaning that relaxing them even slightly would allow for counterexamples. Demonstrating a graph with one fewer edge that lacks the property, as in this problem, is a method of proving that the bound is sharp.

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