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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Name: ____________________   Date: ____________

1.Easy

How many edges |E| does the complete bipartite graph K6,6 have?

K6,6
|E| =
2.Easy

A 3-regular graph has 6 vertices (every vertex has degree 3). How many edges |E| does it have?

n = 6k = 3
|E| =
3.Medium
ABCDEEuler 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| = 5|E| = 6
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
4.Medium

A tree has 4 vertices of degree 3, 3 vertices of degree 5, and all its other vertices are leaves (degree 1). How many leaves L does it have?

4 × deg 33 × deg 5
L =
5.Hard
12115117176ABCDEFG

Use Dijkstra’s algorithm to find the length d(F) of the shortest path from A to F in the weighted graph.

start: Aend: F
d(F) =
6.Hard
148151169911ABCDEF

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

Answer key

  1. 1. |E| = 36
  2. 2. |E| = 9
  3. 3. Euler circuit
  4. 4. L = 15
  5. 5. d(F) = 21
  6. 6. D

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