Graph theory worksheet with answers

Free printable Graph theory worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy

A 4-regular graph has 5 vertices (every vertex has degree 4). How many edges |E| does it have?

n = 5k = 4
|E| =
2.Easy

A simple graph has 5 vertices with degrees 3, 3, 3, 2, 1. How many edges |E| does it have?

deg: 3, 3, 3, 2, 1
|E| =
3.Medium
ABCDEFGEuler circuitEuler path, no circuitneither

Does the connected graph have an Euler circuit, an Euler path that is not a circuit, or neither? Click your answer.

|V| = 7|E| = 12
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
4.Medium

How many different Hamilton circuits H does the complete graph K5 have? Circuits that differ only in the starting vertex or the direction count as the same.

K5
H =
5.Hard
1411414713312ABCDEF

Dijkstra’s algorithm runs from A. Click the vertex that is settled (made permanent) in round 5; A itself is round 1.

start: Around 5
  1. 1A
  2. 2B
  3. 3C
  4. 4D
  5. 5E
  6. 6F
6.Hard
181917144910ABCDE

Find the total weight W of a minimum spanning tree of the weighted graph.

|V| = 5|E| = 7
W =

Answer key

  1. 1. |E| = 10
  2. 2. |E| = 6
  3. 3. Euler circuit
  4. 4. H = 12
  5. 5. F
  6. 6. W = 41

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