March 2018 Detroit Mercy Math Problem

Send solutions to dmmc@udmercy.edu by April 30, 2018 (extended deadline).

The arrangement graph $$A_{n,k}$$ (with $$n>k\geq 1$$) has a vertex set consisting of all possible permutations of $$k$$ elements chosen from the ground set of $$n$$ elements $$\{1,2,\ldots , n\}$$. Two vertices (nodes) $$u$$ and $$v$$ of $$A_{n,k}$$ are adjacent if their corresponding permutations differ in exactly one of the $$k$$ positions.

A hamiltonian cycle in a graph is a closed path through a graph that visits each node exactly once.

And here is the March 2018 problem (PDF)

Find three edge disjoint hamiltonian cycles in $$A_{5,2}$$ shown below.

Figure 1: The graph $$A_{5,2}$$ contains three edge disjoint hamiltonian cycles.

A hint for March 2018 Problem

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