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
ABCDEF

Exactly one vertex of the graph has degree 4. Click it.

deg = 4
  1. 1A
  2. 2B
  3. 3C
  4. 4D
  5. 5E
  6. 6F
2.Easy

A simple graph has 7 vertices with degrees 4, 4, 3, 3, 2, 2, 2. How many edges |E| does it have?

deg: 4, 4, 3, 3, 2, 2, 2
|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| = 5
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
4.Medium

How many different Hamilton circuits H does the complete graph K7 have? Circuits that differ only in the starting vertex or the direction count as the same.

K7
H =
5.Hard
448103141214ABCDEF

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

start: Aend: E
d(E) =
6.Hard
1012181514198ABCDE

Kruskal’s algorithm builds a minimum spanning tree. Click the edge it adds as edge number 2; skipped edges do not count.

k = 2
  1. 1edge AB
  2. 2edge AC
  3. 3edge BC
  4. 4edge BD
  5. 5edge CE
  6. 6edge DE
  7. 7edge BE

Answer key

  1. 1. E
  2. 2. |E| = 10
  3. 3. Euler circuit
  4. 4. H = 360
  5. 5. d(E) = 7
  6. 6. edge AB

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