1. The dodecahedron graph 𝐺 is depicted below:
A. Determine, with justification, whether 𝐺 is Eulerian.
B. Show that 𝐺 is Hamiltonian by finding a Hamilton cycle.
2. Let 𝐻 be the graph depicted to the right: A. Find a 4-coloring of 𝐻.
B. Show that no 3-coloring of 𝐻 exists.
3. The graph 𝑃3×𝑃3 is depicted below. Show that this graph is not Hamiltonian. One approach: Show that any Hamilton path must begin and end at even-numbered vertices. Why does this prevent forming a Hamilton cycle?
4. Find the chromatic polynomial 𝑝𝐺(𝑘)of 𝐺 =𝐶6 and determine whether 𝑘−2 is a factor of 𝑝𝐺(𝑘).
