CPTS 553 Graph Theory Assignment 5 solution

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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).