Names: ___________________
Worksheet #1: Chromatic Numbers
1. Let 𝐿𝑛 be the “line graph” on 𝑛 vertices. What is the chromatic number of 𝐿𝑛 ?
Answer:
2 – you can just alternate the colors.
2. Let 𝐶𝑛 be the “cycle graph” on 𝑛 vertices. What is the chromatic number of 𝐶𝑛 ?
Answer:
If 𝒏 is even, 2 – you can just alternate the colors. If 𝒏 is odd, 3 are required.
3. Let 𝑊𝑛 be the “wheel graph” on 𝑛 vertices. What is the chromatic number of 𝑊𝑛 ?
Answer:
If 𝒏 is even, 3 – you can just alternate the colors on the exterior vertices and
use the third color for the center. If 𝒏 is odd, 4 are required.
4.
Chromatic # = _2_
Chromatic # = _3_
Chromatic # = _4_
(It can’t be <2, since the graph has edges.)
(It can’t be <3, since the graph contains a triangle.)
(It can’t be <4, since the graph contains a copy of 𝐾4 .)
5. Let 𝐾𝑛 denote the complete graph on 𝑛 vertices.
a. What is the chromatic number of 𝐾𝑛 ?
Answer -- 𝒏 : each vertex must be a different color.
b. What is the chromatic number of the graph obtained from 𝐾𝑛 by removing one edge?
Answer -- (𝒏 − 𝟏) : the two vertices not joined by an edge may be given the same color.
c. What is it for the graph obtained from 𝐾𝑛 by removing two edges with a common vertex?
Answer -- (𝒏 − 𝟏) colors are still required. The “common vertex” may be given the same
color as either one of the two vertices it is not connected to but no other color repetition
is allowed.
d. What is it for the graph obtained from 𝐾𝑛 by removing two edges w/o a common vertex?
Answer -- (𝒏 − 𝟐) : There are two pairs of vertices not joined by an edge. Each pair may
be given the same color, so two different colors are repeated.