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

A 2-regular graph has 12 vertices (every vertex has degree 2). How many edges |E| does it have?

n = 12k = 2
|E| =
2.Easy
ABCDEFG

Find the sum of the degrees of all vertices of the graph shown, Σ deg.

|V| = 7
Σ deg =
3.Medium

How many different Hamilton circuits H does the complete graph K8 have? Circuits that differ only in the starting vertex or the direction count as the same.

K8
H =
4.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| = 7
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
5.Hard
8713121012124113ABCDEFG

Dijkstra’s algorithm runs from A. Click the vertex that is settled (made permanent) in round 3; A itself is round 1.

start: Around 3
  1. 1A
  2. 2B
  3. 3C
  4. 4D
  5. 5E
  6. 6F
  7. 7G
6.Hard
823691547ABCDEF

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) =

Answer key

  1. 1. |E| = 12
  2. 2. Σ deg = 18
  3. 3. H = 2520
  4. 4. Euler path, no circuit
  5. 5. D
  6. 6. d(F) = 19

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