Let n ≥3 be an integer. Let Gn be the graph whose vertices...

Let $n \geq 3$ be an integer. Let $G_{n}$ be the graph whose vertices are the $n !$ permutations of $\{1,2, \ldots, n\}$, wherein two permutations are joined by an edge if and only if one can be obtained from the other by the interchange of two numbers (an arbitrary transposition). Deduce from the results of Section $4.1$ that $G_{n}$ has a Hamilton cycle.

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Key Concepts

Hamilton Cycle

A Hamilton cycle is a cycle in a graph that visits every vertex exactly once and returns to the starting vertex. It is a central concept in graph theory and often appears in various problems regarding traversability and routing within graphs.

Cayley Graph

A Cayley graph is a graphical representation of a group with respect to a particular generating set. Its vertices correspond to the elements of the group and its edges represent group operations with generators. This structure is particularly useful in studying the properties of the group and in problems involving Hamilton cycles.

Symmetric Group

The symmetric group on n elements, denoted by S?, consists of all the permutations of n objects. Its rich structure and well-understood set of generators make it a common subject in combinatorial and algebraic problems, such as demonstrating the existence of Hamilton cycles in associated Cayley graphs.

Transposition

A transposition is a simple permutation that swaps two elements while keeping all other elements fixed. Transpositions generate the entire symmetric group, and considering them as generators in the Cayley graph framework leads to interesting graph properties, such as the existence of Hamilton cycles.

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