VIPSolutionsGive step-by-step solution with explanation and final answer:In Exercises 1-8 determine whether the given graph has an Euler circuit. Construct such a circuit when one exists. If no Euler circuit exists, determine whether the graph has an Euler path and construct such a path if one exists. 1. b c 4 2. a b c 1 ¢ 1 i 3.a b 4. a PY ZN , ) ' XK e c
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Give step-by-step solution with explanation and final answer:
In Exercises 1-8 determine whether the given graph has an
Euler circuit. Construct such a circuit when one exists. If
no Euler circuit exists, determine whether the graph has an
Euler path and construct such a path if one exists.
1. b c
4
2. a b c
1
¢ 1 i
3.a b 4. a
PY ZN ,
) ' XK
e c
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Asked by: gvr
Created at: 2025-06-27 03:07:46
Asked by: gvr
Created at: 2025-06-27 03:07:46
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Let’s solve each graph step-by-step to determine if it has an **Euler circuit**, **Euler path**, or neither. Recall:
- **Euler Circuit**: A circuit that uses every edge exactly once and starts and ends at the same vertex. *Exists if and only if every vertex has even degree and the graph is connected.*
- **Euler Path**: A path that uses every edge exactly once, but doesn't have to start and end at the same vertex. *Exists if and only if exactly two vertices have odd degree and the graph is connected.*
---
### 1.
#### **Degrees:**
- a: 3 (odd)
- b: 3 (odd)
- c: 2 (even)
- d: 2 (even)
- e: 2 (even)
#### **Analysis:**
- Odd degree vertices: a, b (2 vertices)
#### **Conclusion:**
- **Euler path exists, but no Euler circuit.**
#### **Euler Path Example:**
Start at **a**, end at **b**:
- a → d → c → b → e → a → b
---
### 2.
#### **Degrees:**
- a: 2 (even)
- b: 4 (even)
- c: 2 (even)
- d: 3 (odd)
- e: 4 (even)
- f: 3 (odd)
- g: 2 (even)
- h: 2 (even)
- i: 2 (even)
#### **Analysis:**
- Odd degree vertices: d, f (2 vertices)
#### **Conclusion:**
- **Euler path exists, but no Euler circuit.**
#### **Euler Path Example:**
Start at **d**, end at **f**:
- d → a → b → e → f → c → b → e → h → g → d → e → f → i → h
(Many possible paths—ensure every edge is used once.)
---
### 3.
#### **Degrees:**
- a: 3 (odd)
- b: 3 (odd)
- c: 3 (odd)
- d: 3 (odd)
- e: 4 (even)
#### **Analysis:**
- Odd degree vertices: a, b, c, d (4 vertices)
#### **Conclusion:**
- **Neither Euler path nor Euler circuit exists** (since more than two vertices have odd degree).
---
### 4.
#### **Degrees:**
- a: 4 (even)
- b: 4 (even)
- c: 2 (even)
- d: 4 (even)
- e: 2 (even)
- f: 2 (even)
#### **Analysis:**
- Odd degree vertices: none
#### **Conclusion:**
- **Euler circuit exists.**
#### **Euler Circuit Example:**
Start at **a**:
- a → b → c → e → f → a → d → f → b → d → c → a
(Many possibilities—ensure every edge is used once and path is a circuit.)
---
## **Final Answers (Summary):**
| Graph | Euler Circuit? | Euler Path? | Example Path/Circuit |
|-------|:--------------:|:-----------:|----------------------|
| 1 | No | Yes | a-d-c-b-e-a-b |
| 2 | No | Yes | d-a-b-e-f-c-b-e-h-g-d-e-f-i-h |
| 3 | No | No | None |
| 4 | Yes | Yes | a-b-c-e-f-a-d-f-b-d-c-a |
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