Graph Coloring Vertex Coloring: A proper k-coloring of a graph G is a labeling f :V (G) → {1,...,k} such that if xy is an edge in G, then f (x) ≠ f (y ) .
A graph G is k-colorable if it has a proper k-coloring.
The chromatic number χ(G) is the minimum k such that G is k-colorable. e.g. Find the chromatic numbers of the following graphs.
Q: Find χ(Kn) (where Kn is a graph on n vertices where each pair of vertices is connected by an edge).
Q: Find χ(Cn)
Let Wn be a graph consisting of a cycle of length n and another vertex v, which is connected to every other vertex. e.g.
W4 W3 W9
Q: Find χ(Wn).
Notation: Let Δ(G) be the vertex of G with the largest degree.
Prove the following statement: For any graph G, χ(G) ≤ Δ(G) + 1.
Find the chromatic number for each of the following graphs. Try to give an argument to show that fewer colors will not suffice. a. b. c.
d. e. f.
Color the 5 Platonic Solids: Coloring Maps Perhaps the most famous problem involves coloring maps.
$1,000,000 Question: Given a (plane) map, what is the least number of colors needed to color the regions so that regions sharing a border are not colored the same? e.g. Determine the # of colors needed to properly color the following maps:
a. b.
c. Color the (flattened versions of the) 5 Platonic Solids below: Some facts about planar graphs, which will help us get the answer to the $1,000,000 question:
1. Recall Euler’s Formula, which said that V + F – E = 2.
2. The maximum number of edges possible in a plane graph on n vertices occurs when all the faces are triangles.
3. In a plane graph, E ≤ 3V – 6
4. In a plane graph, there is at least one vertex with degree less than or equal to 5.
5. If G is a plane graph, χ(G) ≤ 6
History: Check out http://www.math.gatech.edu/~thomas/FC/fourcolor.html
Book on the History: “Four Color Suffices” by Robert Wilson Edge Coloring
A proper k-edge coloring of a graph G is a labeling f : E(G) → {1,...,k } such that edges sharing a vertex receive different colors.
The edge-chromatic number or chromatic index χ'(G) is the minimum k such that G has a proper k-coloring. e.g. a. b. c.
Q: Find χ'(K2n): Q: Find χ'(K2n+1):
Q: Find χ'(C2n): Q: Find χ'(C2n+1):
Q: Find χ'(W2n): Q: Find χ'(W2n+1):
Interesting Theorem (Vizing 1964): If the maximum degree of a vertex in a graph G is Δ(G), then the chromatic index of G is either Δ(G) or Δ(G) + 1.