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 simple graph has 5 vertices with degrees 4, 3, 2, 2, 1. How many edges |E| does it have?

deg: 4, 3, 2, 2, 1
|E| =
2.Easy

A simple graph has 6 vertices with degrees 4, 3, 2, 2, 2, 1. How many edges |E| does it have?

deg: 4, 3, 2, 2, 2, 1
|E| =
3.Medium

A tree has 4 vertices of degree 3, 4 vertices of degree 4, 2 vertices of degree 5, 2 vertices of degree 2, and all its other vertices are leaves (degree 1). How many leaves L does it have?

4 × deg 34 × deg 42 × deg 52 × deg 2
L =
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
13161472011173ABCDEF

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 BD
  4. 4edge CE
  5. 5edge DE
  6. 6edge DF
  7. 7edge EF
  8. 8edge BE
6.Hard
152048129ABCDE

Prim’s algorithm builds a minimum spanning tree starting from vertex B. Click the edge it adds as edge number 4.

start: Bk = 4
  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| = 6
  2. 2. |E| = 7
  3. 3. L = 20
  4. 4. neither
  5. 5. edge AB
  6. 6. edge AB

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