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 forest (a graph without cycles) has 12 vertices and 6 connected components. How many edges |E| does it have?

|V| = 12c = 6
|E| =
2.Easy
ABCDEF

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

|V| = 6
Σ deg =
3.Medium

How many different Hamilton circuits H does the complete graph K4 have? Circuits that differ only in the starting vertex or the direction count as the same.

K4
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| = 6
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
5.Hard
161731028182015ABCDEFG

Prim’s algorithm builds a minimum spanning tree starting from vertex A. Click the edge it adds as edge number 6.

start: Ak = 6
  1. 1edge AB
  2. 2edge AC
  3. 3edge BC
  4. 4edge BD
  5. 5edge CE
  6. 6edge DE
  7. 7edge DF
  8. 8edge EF
  9. 9edge GB
  10. 10edge GD
6.Hard
961148513310ABCDEF

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| = 6
  2. 2. Σ deg = 14
  3. 3. H = 3
  4. 4. Euler circuit
  5. 5. edge GD
  6. 6. F

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