Consider the following graphs over the vertices {a,b,c,d,e,f}
.
Which of them contain Euler paths?
a)
[[a,e],[a,f],[b,d],[b,f],[c,d],[c,f],[d,e],[d,f]]

Contains a Euler path
Contains a Euler Circuit
Contains both
Contains neither.
Answered

b)
[[a,d],[a,e],[b,d],[b,e],[b,f],[c,d],[c,e],[c,f],[d,e],[e,f]]

Contains a Euler path
Contains a Euler Circuit
Contains both
Contains neither.


To determine whether a graph contains an Euler path or an Euler circuit, we need to analyze the degrees of the vertices.

### **Euler Path**
- A graph contains an Euler path if and only if it has exactly **two vertices with an odd degree** (or no odd degree, which would mean it contains an Euler circuit).

### **Euler Circuit**
- A graph contains an Euler circuit if and only if **all vertices have an even degree**.

Let's analyze the two graphs provided.

### **Graph a)** \(\{[a,e],[a,f],[b,d],[b,f],[c,d],[c,f],[d,e],[d,f]\}\)

#### Degrees of the vertices:
- \( \text{deg}(a) = 2 \)  (a is connected to e and f)
- \( \text{deg}(b) = 2 \)  (b is connected to d and f)
- \( \text{deg}(c) = 2 \)  (c is connected to d and f)
- \( \text{deg}(d) = 4 \)  (d is connected to b, c, e, and f)
- \( \text{deg}(e) = 2 \)  (e is connected to a and d)
- \( \text{deg}(f) = 4 \)  (f is connected to a, b, c, and d)

#### Analysis:
- All vertices have even degrees.

**Conclusion for Graph a:**
- Since all vertices have even degrees, **Graph a** contains both an **Euler circuit** and an **Euler path**.

### **Graph b)** \(\{[a,d],[a,e],[b,d],[b,e],[b,f],[c,d],[c,e],[c,f],[d,e],[e,f]\}\)

#### Degrees of the vertices:
- \( \text{deg}(a) = 2 \)  (a is connected to d and e)
- \( \text{deg}(b) = 3 \)  (b is connected to d, e, and f)
- \( \text{deg}(c) = 3 \)  (c is connected to d, e, and f)
- \( \text{deg}(d) = 4 \)  (d is connected to a, b, c, and e)
- \( \text{deg}(e) = 4 \)  (e is connected to a, b, c, and d)
- \( \text{deg}(f) = 2 \)  (f is connected to b and c)

#### Analysis:
- Vertices \(b\) and \(c\) have odd degrees (3 each).

**Conclusion for Graph b:**
- Since there are exactly two vertices with odd degrees, **Graph b** contains an **Euler path** but does not contain an Euler circuit.

### **Final Answer**:
- **Graph a**: Contains both an Euler path and an Euler circuit.
- **Graph b**: Contains an Euler path only.

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:
1. How would the result change if one edge was added to or removed from Graph a?
2. What happens if a graph has more than two vertices with odd degrees?
3. How can you determine if a graph is connected, and why is this important for Euler paths?
4. What are the practical applications of Euler paths and circuits in real life?
5. How would the presence of isolated vertices affect the existence of Euler paths and circuits?

**Tip:** Remember that the sum of all vertex degrees in a graph is always even, as each edge contributes exactly 2 to the sum (one for each endpoint).

Explore Euler paths and circuits in graph theory with detailed analysis of graphs containing vertices {a, b, c, d, e, f}. Understand when a graph contains an Euler path, an Euler circuit, both, or neither based on vertex degrees.

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