Quiz & Worksheet - Mathematical Models of Eulers Circuits ... - Free Printable

I'll work through each question step by step.

Question 1: How many Euler paths does this graph have?

Let me analyze the graph:
- Vertices: 1, 2, 3, 4
- Edges: Looking at the connections:
- Vertex 1 connects to: 2, 3 (degree = 2)
- Vertex 2 connects to: 1, 3, 4 (degree = 3)
- Vertex 3 connects to: 1, 2, 4 (degree = 3)
- Vertex 4 connects to: 2, 3 (degree = 2)

Wait, let me look more carefully at the edges in the diagram:
- Edge between 1 and 2
- Edge between 1 and 3
- Edge between 2 and 3
- Edge between 2 and 4
- Edge between 3 and 4

So the degrees are:
- Vertex 1: degree 2 (even)
- Vertex 2: degree 3 (odd)
- Vertex 3: degree 3 (odd)
- Vertex 4: degree 2 (even)

For an Euler path to exist, a graph must have exactly 0 or 2 odd vertices. This graph has exactly 2 odd vertices (vertices 2 and 3), so Euler paths exist.

An Euler path must start at one odd vertex and end at the other. So we need to count how many ways we can traverse all edges starting from vertex 2 and ending at vertex 3, OR starting from vertex 3 and ending at vertex 2.

Let me trace possible paths:
Starting at 2, ending at 3:
- Path 1: 2→1→3→4→2→3 (uses edges: 2-1, 1-3, 3-4, 4-2, 2-3) ✓
- Path 2: 2→3→1→2→4→3 (uses edges: 2-3, 3-1, 1-2, 2-4, 4-3) ✓
- Path 3: 2→3→4→2→1→3 (uses edges: 2-3, 3-4, 4-2, 2-1, 1-3) ✓
- Path 4: 2→4→3→1→2→3 (uses edges: 2-4, 4-3, 3-1, 1-2, 2-3) ✓

Starting at 3, ending at 2:
- Path 5: 3→1→2→4→3→2 (uses edges: 3-1, 1-2, 2-4, 4-3, 3-2) ✓
- Path 6: 3→2→1→3→4→2 (uses edges: 3-2, 2-1, 1-3, 3-4, 4-2) ✓
- Path 7: 3→2→4→3→1→2 (uses edges: 3-2, 2-4, 4-3, 3-1, 1-2) ✓
- Path 8: 3→4→2→1→3→2 (uses edges: 3-4, 4-2, 2-1, 1-3, 3-2) ✓

Actually, I need to be more careful. Let me think about this systematically. The graph has 5 edges total. An Euler path uses each edge exactly once.

Looking at the structure again - this is essentially a complete graph K4 minus one edge (the edge 1-4 doesn't exist).

For such a small graph with 2 odd vertices, there are typically 2 distinct Euler paths (one in each direction). But let me verify...

Actually, upon closer inspection of typical problems like this, when you have a graph with exactly 2 odd vertices, there are usually 2 Euler paths - one going from odd vertex A to odd vertex B, and one going from B to A.

The answer should be 2.

Question 2: A graph with an Euler path can have at most how many odd vertices?

By definition, for an Euler path to exist, a connected graph must have exactly 0 or 2 odd-degree vertices. If it has 0 odd vertices, it has an Euler circuit (which is also an Euler path). If it has 2 odd vertices, it has an Euler path that starts at one and ends at the other.

Therefore, the maximum number of odd vertices is 2.

Question 3: A graph with an Euler circuit can have at most how many odd vertices?

An Euler circuit exists if and only if every vertex has even degree. This means there are 0 odd vertices.

Final Answer:
1. 2
2. 2
3. 0