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

A forest (a graph without cycles) has 7 vertices and 3 connected components. How many edges |E| does it have?

|V| = 7c = 3
|E| =
2.Easy

A 4-regular graph has 9 vertices (every vertex has degree 4). How many edges |E| does it have?

n = 9k = 4
|E| =
3.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| = 7
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
4.Medium

A connected graph has 17 vertices and 23 edges. How many edges k must be removed to leave a spanning tree?

|V| = 17|E| = 23
k =
5.Hard
4811519813ABCDEF

Dijkstra’s algorithm runs from A. Click the vertex that is settled (made permanent) in round 4; A itself is round 1.

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

How many closed walks of length 2 start and end at C? A walk may repeat vertices and edges. Call the number N.

M = adjacency matrixL = 2
N =

Answer key

  1. 1. |E| = 4
  2. 2. |E| = 18
  3. 3. neither
  4. 4. k = 7
  5. 5. E
  6. 6. N = 3

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