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

How many edges |E| does the complete bipartite graph K4,6 have?

K4,6
|E| =
2.Easy

A simple graph has 6 vertices with degrees 4, 2, 2, 2, 2, 2. How many edges |E| does it have?

deg: 4, 2, 2, 2, 2, 2
|E| =
3.Medium

How many different Hamilton circuits H does the complete graph K6 have? Circuits that differ only in the starting vertex or the direction count as the same.

K6
H =
4.Medium
ABCDEFEuler 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| = 6|E| = 8
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
5.Hard
3181261311220ABCDEF

Kruskal’s algorithm builds a minimum spanning tree. Click the edge it adds as edge number 4; skipped edges do not count.

k = 4
  1. 1edge AB
  2. 2edge AC
  3. 3edge BC
  4. 4edge BD
  5. 5edge CE
  6. 6edge DE
  7. 7edge DF
  8. 8edge BE
6.Hard
1182041512143287ABCDEFG

Prim’s algorithm builds a minimum spanning tree starting from vertex D. Click the edge it adds as edge number 2.

start: Dk = 2
  1. 1edge AB
  2. 2edge AC
  3. 3edge BC
  4. 4edge BD
  5. 5edge DE
  6. 6edge DF
  7. 7edge EF
  8. 8edge GB
  9. 9edge GC
  10. 10edge GD
  11. 11edge GE

Answer key

  1. 1. |E| = 24
  2. 2. |E| = 7
  3. 3. H = 60
  4. 4. Euler path, no circuit
  5. 5. edge DE
  6. 6. edge AB

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