Database System Concepts
Database System Concepts
7th Edition
ISBN: 9780078022159
Author: Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher: McGraw-Hill Education
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The class I'm taking is computer science discrete structures. 

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**Educational Content: Euler Paths and Circuits**

**Problem Statement:**

Which of the following graphs contain an Euler path? Which contain an Euler circuit?

(a) \( K_4 \)  
(b) \( K_5 \)  
(c) \( K_{5,7} \)   
(d) \( K_{2,7} \)  
(e) \( C_7 \)  
(f) \( P_7 \)  

**Definitions:**

- **Euler Path:** A trail in a graph that visits every edge exactly once. An Euler path exists if a graph has exactly zero or two vertices of odd degree.
  
- **Euler Circuit:** A circuit in a graph that visits every edge exactly once and returns to the starting vertex. An Euler circuit exists if all vertices in the graph have even degrees.

**Graph Notations:**

- **\( K_n \):** A complete graph on \( n \) vertices, where there is an edge between every pair of vertices.

- **\( K_{m,n} \):** A complete bipartite graph where vertices are divided into two disjoint sets of \( m \) and \( n \) vertices, and every vertex from one set is connected to every vertex in the other set.

- **\( C_n \):** A cycle graph on \( n \) vertices, forming a single loop.

- **\( P_n \):** A path graph on \( n \) vertices, forming a single straight line without any loops.
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Transcribed Image Text:**Educational Content: Euler Paths and Circuits** **Problem Statement:** Which of the following graphs contain an Euler path? Which contain an Euler circuit? (a) \( K_4 \) (b) \( K_5 \) (c) \( K_{5,7} \) (d) \( K_{2,7} \) (e) \( C_7 \) (f) \( P_7 \) **Definitions:** - **Euler Path:** A trail in a graph that visits every edge exactly once. An Euler path exists if a graph has exactly zero or two vertices of odd degree. - **Euler Circuit:** A circuit in a graph that visits every edge exactly once and returns to the starting vertex. An Euler circuit exists if all vertices in the graph have even degrees. **Graph Notations:** - **\( K_n \):** A complete graph on \( n \) vertices, where there is an edge between every pair of vertices. - **\( K_{m,n} \):** A complete bipartite graph where vertices are divided into two disjoint sets of \( m \) and \( n \) vertices, and every vertex from one set is connected to every vertex in the other set. - **\( C_n \):** A cycle graph on \( n \) vertices, forming a single loop. - **\( P_n \):** A path graph on \( n \) vertices, forming a single straight line without any loops.
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