CS 594 Graph Theory, Spring 2014 Homework 5
1. Using initial graph P2, how many edges are in a graph with chromatic number n created using
Mycielski’s construction?
2. Prove or disprove: Kn has the fewest possible edges of any graph with chromatic number n.
3. Prove or disprove: Every graph has a minimum coloring in which all vertices in a maximum
independent set are the same color.
4. Prove or disprove: The chromatic number of G is the maximum of the chromatic numbers of
the blocks of G.
5. The Southeastern Conference consists of universities from Alabama, Arkansas, Florida,
Georgia, Kentucky, Louisiana, Mississippi, Missouri, South Carolina, Tennessee, and Texas.
Draw a graph GSEC where the vertices are states and the edges are borders between states. What
is χ(GSEC )? Will a greedy coloring always color GSEC with χ(GSEC ) colors? If not, give an
example.
6. Prove or disprove: If χ(G) = Δ(G), then χ( ) < Δ( ).
7. Prove that if G is bipartite, then χ( ) is the size of the maximum clique in .